A particle moves with speed 0.800c relative to a frame S'', which itself moves at 0.800c relative to frame S'. Frame S' moves at 0.800c relative to frame S.
(a) The speed of the particle relative to frame S' is calculated to be 0.976c by applying the inverse velocity transformation equation.
(b) The speed of the particle relative to the lab frame S is calculated to be 0.997c by applying the inverse velocity transformation equation to express the speed in terms of the speed relative to S'.
This document discusses Maxwell's equations and electromagnetic waves through conceptual problems and examples.
Some key points:
1) Maxwell's equations apply to both time-independent and time-dependent electric and magnetic fields. The electromagnetic wave equation can be derived from Maxwell's equations.
2) Electromagnetic waves are transverse waves where the electric and magnetic fields oscillate perpendicular to the direction of propagation.
3) The momentum of an electromagnetic wave depends on its intensity, so waves of equal intensity have equal momentum regardless of frequency.
4) Radiation pressure from sunlight was determined to be causing changes to the orbit of one of the first U.S. satellites, something not accounted for in its design. Estimates
- Carlos Ragone fell 500 feet into snow after his anchor gave way while mountain climbing. The snow broke his fall, creating a 4 foot deep hole.
- Assuming constant acceleration during his impact with the snow, the estimated average acceleration was about 125g as he slowed to a stop within the 4 foot deep hole.
- A runner ran 2.5 km in 9 minutes, then walked back to the starting point over 30 minutes. Her average velocity was 5.2 km/hr for running, -0.83 km/hr for walking, and 0 km/hr for the total trip. Her average speed for the entire trip was 1.3 km/hr.
Three events occur:
1) An airplane carrying an atomic clock flies back and forth for 15 hours at an average speed of 140 m/s.
2) The time on the clock is compared to an atomic clock kept on the ground.
3) Using time dilation, the elapsed time on the ground clock is calculated to be 9.5 nanoseconds longer than the time on the airplane clock.
This document contains a mathematics enrichment module on representing the Earth as a sphere. It includes 6 problems involving calculating distances, times, speeds and locations on the Earth's surface given latitudes and longitudes. The problems require applying concepts like the ratio of distances to points via the North and South poles, converting between angular and linear distances, and solving related rate problems involving travel along parallels of latitude or meridians of longitude.
The document contains sample questions and solutions for understanding concepts related to distance-time graphs and speed-time graphs. It introduces key ideas such as calculating speed from the gradient of a distance-time graph, calculating average speed and acceleration from areas under graphs, and using graphs to solve word problems about distance, speed, and time for moving objects. Several practice exercises with multiple choice and short answer questions are provided to help students apply these graph-based concepts.
Module 13 Gradient And Area Under A Graphguestcc333c
1) The document provides examples and questions related to calculating gradient, area under graphs, speed, velocity, and distance from speed-time and distance-time graphs.
2) It includes 10 multi-part questions testing concepts like calculating rate of change of speed, uniform speed, total distance, meeting time, and average speed.
3) Detailed step-by-step answers are provided for each question at the end to demonstrate how to apply the concepts to calculate the requested values.
This chapter discusses simple harmonic motion (SHM). SHM is defined as periodic motion where the acceleration is directly proportional to and opposite of the displacement from equilibrium. The key equations of SHM are introduced, including the displacement equation x = A sin(ωt + φ) and equations for velocity, acceleration, kinetic energy, and potential energy using angular frequency ω. Examples of SHM include a simple pendulum and spring oscillations. Exercises are provided to apply the kinematic equations of SHM.
Modul 4-balok menganjur diatas dua perletakanMOSES HADUN
The document discusses beam structures that project over two supports (overhangs) and are loaded in various ways. It provides 5 examples of overhang beams with different load configurations: 1) a centered point load, 2) a uniformly distributed load, 3) a combination of uniform and point loads, 4) centered point loads on both sides, and 5) a uniform load on both sides. For each example, it solves for the support reactions, internal shear forces, and bending moments. The goal is for students to understand the internal forces in overhang beams under different loading conditions and be able to calculate the reactions, shear forces, and bending moments.
This document discusses Maxwell's equations and electromagnetic waves through conceptual problems and examples.
Some key points:
1) Maxwell's equations apply to both time-independent and time-dependent electric and magnetic fields. The electromagnetic wave equation can be derived from Maxwell's equations.
2) Electromagnetic waves are transverse waves where the electric and magnetic fields oscillate perpendicular to the direction of propagation.
3) The momentum of an electromagnetic wave depends on its intensity, so waves of equal intensity have equal momentum regardless of frequency.
4) Radiation pressure from sunlight was determined to be causing changes to the orbit of one of the first U.S. satellites, something not accounted for in its design. Estimates
- Carlos Ragone fell 500 feet into snow after his anchor gave way while mountain climbing. The snow broke his fall, creating a 4 foot deep hole.
- Assuming constant acceleration during his impact with the snow, the estimated average acceleration was about 125g as he slowed to a stop within the 4 foot deep hole.
- A runner ran 2.5 km in 9 minutes, then walked back to the starting point over 30 minutes. Her average velocity was 5.2 km/hr for running, -0.83 km/hr for walking, and 0 km/hr for the total trip. Her average speed for the entire trip was 1.3 km/hr.
Three events occur:
1) An airplane carrying an atomic clock flies back and forth for 15 hours at an average speed of 140 m/s.
2) The time on the clock is compared to an atomic clock kept on the ground.
3) Using time dilation, the elapsed time on the ground clock is calculated to be 9.5 nanoseconds longer than the time on the airplane clock.
This document contains a mathematics enrichment module on representing the Earth as a sphere. It includes 6 problems involving calculating distances, times, speeds and locations on the Earth's surface given latitudes and longitudes. The problems require applying concepts like the ratio of distances to points via the North and South poles, converting between angular and linear distances, and solving related rate problems involving travel along parallels of latitude or meridians of longitude.
The document contains sample questions and solutions for understanding concepts related to distance-time graphs and speed-time graphs. It introduces key ideas such as calculating speed from the gradient of a distance-time graph, calculating average speed and acceleration from areas under graphs, and using graphs to solve word problems about distance, speed, and time for moving objects. Several practice exercises with multiple choice and short answer questions are provided to help students apply these graph-based concepts.
Module 13 Gradient And Area Under A Graphguestcc333c
1) The document provides examples and questions related to calculating gradient, area under graphs, speed, velocity, and distance from speed-time and distance-time graphs.
2) It includes 10 multi-part questions testing concepts like calculating rate of change of speed, uniform speed, total distance, meeting time, and average speed.
3) Detailed step-by-step answers are provided for each question at the end to demonstrate how to apply the concepts to calculate the requested values.
This chapter discusses simple harmonic motion (SHM). SHM is defined as periodic motion where the acceleration is directly proportional to and opposite of the displacement from equilibrium. The key equations of SHM are introduced, including the displacement equation x = A sin(ωt + φ) and equations for velocity, acceleration, kinetic energy, and potential energy using angular frequency ω. Examples of SHM include a simple pendulum and spring oscillations. Exercises are provided to apply the kinematic equations of SHM.
Modul 4-balok menganjur diatas dua perletakanMOSES HADUN
The document discusses beam structures that project over two supports (overhangs) and are loaded in various ways. It provides 5 examples of overhang beams with different load configurations: 1) a centered point load, 2) a uniformly distributed load, 3) a combination of uniform and point loads, 4) centered point loads on both sides, and 5) a uniform load on both sides. For each example, it solves for the support reactions, internal shear forces, and bending moments. The goal is for students to understand the internal forces in overhang beams under different loading conditions and be able to calculate the reactions, shear forces, and bending moments.
Serway, raymond a physics for scientists and engineers (6e) solutionsTatiani Andressa
This document contains a chapter outline and sample questions and solutions for a physics and measurement chapter. The chapter outline lists topics like standards of length, mass and time, density and atomic mass, and dimensional analysis. The questions and solutions provide examples of calculations involving converting between units, determining densities, and applying dimensional analysis.
The document discusses the Doppler effect and how it is used to measure motion. It begins by defining the Doppler shift as the difference between the frequency of waves emitted by a source and the frequency received by an observer due to their relative motion. It then provides examples of the Doppler effect for sound and light waves. The rest of the document discusses Doppler shift derivation, geometry, frequency extraction, pulsed Doppler, and different Doppler sensors. It provides illustrations of concepts like Doppler shift for different source and receiver velocities, Doppler geometry, instrumentation for detecting Doppler shifts, and spectrograms from Doppler ultrasound and radar.
Four experiments were conducted using a paint can hanging from a spring. In the first experiment, the paint can oscillated purely vertically, and PCA isolated this behavior in a single principal component, capturing 95% of the variance. When noise was added by shaking the cameras in the second experiment, PCA was still able to isolate the oscillatory behavior but with less accuracy. In experiments three and four where the paint can moved in both vertical and horizontal directions, PCA extracted the multidimensional behavior with the expected rank and reasonable accuracy.
Multi-Fidelity Optimization of a High Speed, Foil-Assisted Catamaran for Low ...Kellen Betts
This document discusses a multi-fidelity optimization of a high-speed, foil-assisted catamaran design for low wake in Puget Sound. It describes the motivation and objectives to reduce vessel wake through hull geometry optimization and lifting surfaces. It outlines the computational models, including a low-fidelity potential flow model and high-fidelity URANS model. It also discusses the multi-objective global optimization approach, including parameterization methods, interpolation methods, and optimization algorithms. The document notes that results will include the final optimized design and sea trial validation.
Accelerating Dynamic Time Warping Subsequence Search with GPUDavide Nardone
Many time series data mining problems require
subsequence similarity search as a subroutine. While this can
be performed with any distance measure, and dozens of
distance measures have been proposed in the last decade, there
is increasing evidence that Dynamic Time Warping (DTW) is
the best measure across a wide range of domains. Given
DTW’s usefulness and ubiquity, there has been a large
community-wide effort to mitigate its relative lethargy.
Proposed speedup techniques include early abandoning
strategies, lower-bound based pruning, indexing and
embedding. In this work we argue that we are now close to
exhausting all possible speedup from software, and that we
must turn to hardware-based solutions if we are to tackle the
many problems that are currently untenable even with stateof-
the-art algorithms running on high-end desktops. With this
motivation, we investigate both GPU (Graphics Processing
Unit) and FPGA (Field Programmable Gate Array) based
acceleration of subsequence similarity search under the DTW
measure. As we shall show, our novel algorithms allow GPUs,
which are typically bundled with standard desktops, to achieve
two orders of magnitude speedup. For problem domains which
require even greater scale up, we show that FPGAs costing just
a few thousand dollars can be used to produce four orders of
magnitude speedup. We conduct detailed case studies on the
classification of astronomical observations and similarity
search in commercial agriculture, and demonstrate that our
ideas allow us to tackle problems that would be simply
untenable otherwise.
1) A spacecraft measures the redshift of photons emitted from the surface of a star to determine the star's mass M and radius R. As it approaches the star, it measures the velocity needed for resonant absorption of photons by He+ ions.
2) The experimental data gives the velocity needed for resonance at different distances from the star. This data is plotted to determine M and R graphically.
3) In addition to gravitational redshift, the emitted photons will experience a small relativistic frequency shift due to the recoil of the emitting atom. This effect is much smaller than the gravitational redshift.
RADAR - RAdio Detection And Ranging
This is the Part 1 of 2 of RADAR Introduction.
For comments please contact me at solo.hermelin@gmail.com.
For more presentation on different subjects visit my website at http://www.solohermelin.com.
Part of the Figures were not properly downloaded. I recommend viewing the presentation on my website under RADAR Folder.
This document discusses the application of synchronized phasor measurement in real-time wide-area monitoring. It provides an overview of phasor and synchrophasor measurement techniques using Fourier transforms. It also discusses power system stability and transient stability. The document demonstrates the monitoring of a multi-machine system using synchronized phasor measurements by simulating various fault conditions and load changes on a 3-machine, 9-bus system and observing the results with a phasor measurement unit to analyze stability.
This document contains physics formulae related to mechanics, thermodynamics, electromagnetism, optics, modern physics and more. Some key formulae include:
Density = mass / volume, Force = rate of change of momentum, Kinetic energy = (1/2)mv^2, Ohm's law: V=IR, Index of refraction n=c/v, Half life of radioactive element t1/2=ln(2)/λ, Bohr's model: L=nh/2π.
The document provides an overview of key physics equations and concepts for Form 4 students, including equations for relative deviation, prefixes, units for area and volume, equations for average speed, velocity, acceleration, momentum, Newton's laws of motion, and impulse. Key terms are defined for important concepts like displacement, time, mass, force, and velocity. Formulas are presented for calculations involving these fundamental physics quantities and relationships.
The document summarizes the heavy-ion physics program using the Large Hadron Collider (LHC) detectors. It discusses probing novel regimes of high density saturated gluon distributions and qualitatively new physics. Key observables include jet quenching, quarkonia suppression, and heavy flavor modification to study the quark-gluon plasma produced in Pb-Pb collisions. The ALICE, ATLAS and CMS experiments are well-suited to measure bulk properties and select hard probes over a wide momentum range.
1) The document describes projects to obtain timing figures for orbit tracking filters using different satellite data window lengths.
2) Timing runs show the Gauss-Newton filter takes 0.484 seconds while the Kalman and Swerling filters take around 0.5 seconds for a 500,000 data point run.
3) Results are shown for the Molniya and GPS-10 satellites using window lengths of 350, 650, and 1250 seconds, demonstrating that doubling the window length improves accuracy by a factor of around 2.8.
Aerodynamics Part I of 3 describes aerodynamics of wings and bodies in subsonic flight.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
This document provides solutions to 24 problems in special relativity from an undergraduate physics textbook. It was created by Charles Asman, Adam Monahan and Malcolm McMillan at the University of British Columbia for their physics students. The problems cover various topics in special relativity including time dilation, length contraction, relativistic Doppler shift, and Lorentz transformations. Standard frames of reference and equations are defined. Detailed step-by-step solutions are provided for each problem.
The document discusses response spectra, which are plots of the maximum response of single-degree-of-freedom oscillators versus their natural period when subjected to a specific ground motion. Response spectra allow characterization of earthquake ground motions and are commonly used in earthquake-resistant design. The analysis procedure using response spectra involves obtaining the spectral acceleration from the response spectrum curve based on an oscillator's period and calculating the maximum displacement, base shear, and overturning moment. This procedure can be extended to multi-degree-of-freedom structures. An example is provided to demonstrate using a response spectrum to determine interstory drifts and story shears in a two-story building.
This document contains lecture notes on ultrasound physics and imaging. It covers fundamental acoustic topics like wave equations, diffraction theory, and the Doppler effect. It discusses how ultrasound waves propagate and interact with tissue, including nonlinear effects. Image formation concepts are introduced, like signal modeling, processing, and statistics. Specific lectures will cover acoustic interactions with tissue, Doppler ultrasound, and special topics. The document provides equations and diagrams to explain longitudinal and shear waves, radiation of ultrasound from apertures, the near field, and focused transducers.
This document summarizes a presentation on using the 4th order Runge-Kutta method to model the trajectory of projectiles. It discusses ordinary differential equations, introduces the RK4 method, and applies it to model the trajectory of unguided missiles accounting for gravity and drag forces, as well as guided self-propelled missiles where thrust is also considered. Input parameters for a simulation of a North Korean missile are provided and the optimal range is determined. Limitations of the model and references are also noted.
Please show the whole code... Im very confused. Matlab code for ea.pdfarshiartpalace
Please show the whole code... I\'m very confused. Matlab code for earth orbit Using a software
package such as MATLAB, create a plot of an Earth orbit with a = 31, 800 km and e = 0.6 for 0
lessthanorequalto theta lessthanorequalto 2 pi. The origin of the plot should be Earth. Perigee
should be in the direction of the positive x axis (therta = 0). On this plot, you should label (by
hand) the semimajor axis, semiminor axis, radius of perigee, radius of apogee, and the
semiparameter (semilatus rectum) of the ellipse.
Solution
---------------------------------------------------------------------------------------------------------------------
-------------------------------------------------
Elliptic orbit:
%periapsis
rp = 500e3 + getradius(\'earth\');
%eccentricity
ec = .7;
%semimajor axis
a = getsemimajoraxis(rp,ec);
%apoapsis
ra = 2*a-rp;
%true anomaly
theta = [0:.1:360];
orbit = ellipse(theta,a,ec,\'earth\');
figure(1), plot(orbit.x,orbit.y); axis equal
Re = getradius(\'earth\');
x = Re*cosd(theta);
y = Re*sind(theta);
hold on
plot(x,y,\'color\',[0 1 1])
hold off
figure(2), plot(theta,orbit.V/1000)
xlabel(\'\\theta\'), ylabel(\'V (km/s)\')
%current time
c = clock;
%reset hour and second
c(4) = 12; %hour
c(5) = 0; %minute
c(6) = orbit.tau; %seconds
%ellapsed time
datetime(c)
%returns 14-Feb-2015 21:34:54, so it\'s about 9.5 hours later still on
%Monday
2
ans =
22-Feb-2015 21:34:54
-----------Elliptic orbit
%periapsis
rp = 500e3 + getradius(\'earth\');
%eccentricity
ec = .7;
%semimajor axis
a = getsemimajoraxis(rp,ec);
%apoapsis
ra = 2*a-rp;
%true anomaly
theta = [0:.1:360];
orbit = ellipse(theta,a,ec,\'earth\');
figure(1), plot(orbit.x,orbit.y); axis equal
Re = getradius(\'earth\');
x = Re*cosd(theta);
y = Re*sind(theta);
hold on
plot(x,y,\'color\',[0 1 1])
hold off
figure(2), plot(theta,orbit.V/1000)
xlabel(\'\\theta\'), ylabel(\'V (km/s)\')
%current time
c = clock;
%reset hour and second
c(4) = 12; %hour
c(5) = 0; %minute
c(6) = orbit.tau; %seconds
%ellapsed time
datetime(c)
%returns 14-Feb-2015 21:34:54, so it\'s about 9.5 hours later still on
%Monday
2
ans =
22-Feb-2015 21:34:54
3
The circular orbits at perigee and apogee are
7.6167 and 3.1997 km/s, respectively.
-----Parabolic orbit:
Sphere of influence is actually a chapter 5 concept, but is the distance at which the gravitational
forces
between two objects is equal. Thus,
While an approximation, if we just estimate this as the point between the sun and the earth
where
these forces balance, then if the orbital radius of Earth is , then
, and thus solution for r_E in gives
r_soi = getsphereofinfluence(\'earth\',\'sun\');
display([\'Radius from center of earth to sphere of influence is approximately \' num2str(r_soi)]);
%perifocal distance
rp = 500e3 + getradius(\'earth\');
%eccentricty
ec = 1; %always so for a a parabola
theta = -180:1:180;
orbit = parabola(theta,rp,\'earth\');
figure(3), plot(orbit.x,orbit.y); axis equal
Re = getradius(\'earth\');
x = Re*cosd(theta);
y = Re*sind(theta);
.
1. The document discusses causality and relativity of simultaneity using spacetime diagrams. It explains that nothing can travel faster than the speed of light based on analyses of pole-barn paradox scenarios and Lorentz transformations.
2. Key concepts from special relativity are reviewed, including Lorentz transformations, 4-vectors like 4-displacement and 4-velocity, proper time, rest mass, 4-momentum, and conservation of 4-momentum. Collisions are also discussed in relation to conservation of 4-momentum.
3. Causality is maintained in special relativity because the order of events cannot be reversed between reference frames if they are causally connected within or on each other's light cones. Faster-
This document outlines the key aspects of using particle-based Monte Carlo simulations to solve the Boltzmann transport equation (BTE) for modeling semiconductor device transport. It describes how the BTE can be solved by decomposing carrier transport into free flight periods between scattering events. Random flight times are generated from the probability distribution of scattering rates. After each free flight, a scattering mechanism is chosen randomly based on its probability. New carrier momentum and energy are determined after each scattering event to model transport.
Serway, raymond a physics for scientists and engineers (6e) solutionsTatiani Andressa
This document contains a chapter outline and sample questions and solutions for a physics and measurement chapter. The chapter outline lists topics like standards of length, mass and time, density and atomic mass, and dimensional analysis. The questions and solutions provide examples of calculations involving converting between units, determining densities, and applying dimensional analysis.
The document discusses the Doppler effect and how it is used to measure motion. It begins by defining the Doppler shift as the difference between the frequency of waves emitted by a source and the frequency received by an observer due to their relative motion. It then provides examples of the Doppler effect for sound and light waves. The rest of the document discusses Doppler shift derivation, geometry, frequency extraction, pulsed Doppler, and different Doppler sensors. It provides illustrations of concepts like Doppler shift for different source and receiver velocities, Doppler geometry, instrumentation for detecting Doppler shifts, and spectrograms from Doppler ultrasound and radar.
Four experiments were conducted using a paint can hanging from a spring. In the first experiment, the paint can oscillated purely vertically, and PCA isolated this behavior in a single principal component, capturing 95% of the variance. When noise was added by shaking the cameras in the second experiment, PCA was still able to isolate the oscillatory behavior but with less accuracy. In experiments three and four where the paint can moved in both vertical and horizontal directions, PCA extracted the multidimensional behavior with the expected rank and reasonable accuracy.
Multi-Fidelity Optimization of a High Speed, Foil-Assisted Catamaran for Low ...Kellen Betts
This document discusses a multi-fidelity optimization of a high-speed, foil-assisted catamaran design for low wake in Puget Sound. It describes the motivation and objectives to reduce vessel wake through hull geometry optimization and lifting surfaces. It outlines the computational models, including a low-fidelity potential flow model and high-fidelity URANS model. It also discusses the multi-objective global optimization approach, including parameterization methods, interpolation methods, and optimization algorithms. The document notes that results will include the final optimized design and sea trial validation.
Accelerating Dynamic Time Warping Subsequence Search with GPUDavide Nardone
Many time series data mining problems require
subsequence similarity search as a subroutine. While this can
be performed with any distance measure, and dozens of
distance measures have been proposed in the last decade, there
is increasing evidence that Dynamic Time Warping (DTW) is
the best measure across a wide range of domains. Given
DTW’s usefulness and ubiquity, there has been a large
community-wide effort to mitigate its relative lethargy.
Proposed speedup techniques include early abandoning
strategies, lower-bound based pruning, indexing and
embedding. In this work we argue that we are now close to
exhausting all possible speedup from software, and that we
must turn to hardware-based solutions if we are to tackle the
many problems that are currently untenable even with stateof-
the-art algorithms running on high-end desktops. With this
motivation, we investigate both GPU (Graphics Processing
Unit) and FPGA (Field Programmable Gate Array) based
acceleration of subsequence similarity search under the DTW
measure. As we shall show, our novel algorithms allow GPUs,
which are typically bundled with standard desktops, to achieve
two orders of magnitude speedup. For problem domains which
require even greater scale up, we show that FPGAs costing just
a few thousand dollars can be used to produce four orders of
magnitude speedup. We conduct detailed case studies on the
classification of astronomical observations and similarity
search in commercial agriculture, and demonstrate that our
ideas allow us to tackle problems that would be simply
untenable otherwise.
1) A spacecraft measures the redshift of photons emitted from the surface of a star to determine the star's mass M and radius R. As it approaches the star, it measures the velocity needed for resonant absorption of photons by He+ ions.
2) The experimental data gives the velocity needed for resonance at different distances from the star. This data is plotted to determine M and R graphically.
3) In addition to gravitational redshift, the emitted photons will experience a small relativistic frequency shift due to the recoil of the emitting atom. This effect is much smaller than the gravitational redshift.
RADAR - RAdio Detection And Ranging
This is the Part 1 of 2 of RADAR Introduction.
For comments please contact me at solo.hermelin@gmail.com.
For more presentation on different subjects visit my website at http://www.solohermelin.com.
Part of the Figures were not properly downloaded. I recommend viewing the presentation on my website under RADAR Folder.
This document discusses the application of synchronized phasor measurement in real-time wide-area monitoring. It provides an overview of phasor and synchrophasor measurement techniques using Fourier transforms. It also discusses power system stability and transient stability. The document demonstrates the monitoring of a multi-machine system using synchronized phasor measurements by simulating various fault conditions and load changes on a 3-machine, 9-bus system and observing the results with a phasor measurement unit to analyze stability.
This document contains physics formulae related to mechanics, thermodynamics, electromagnetism, optics, modern physics and more. Some key formulae include:
Density = mass / volume, Force = rate of change of momentum, Kinetic energy = (1/2)mv^2, Ohm's law: V=IR, Index of refraction n=c/v, Half life of radioactive element t1/2=ln(2)/λ, Bohr's model: L=nh/2π.
The document provides an overview of key physics equations and concepts for Form 4 students, including equations for relative deviation, prefixes, units for area and volume, equations for average speed, velocity, acceleration, momentum, Newton's laws of motion, and impulse. Key terms are defined for important concepts like displacement, time, mass, force, and velocity. Formulas are presented for calculations involving these fundamental physics quantities and relationships.
The document summarizes the heavy-ion physics program using the Large Hadron Collider (LHC) detectors. It discusses probing novel regimes of high density saturated gluon distributions and qualitatively new physics. Key observables include jet quenching, quarkonia suppression, and heavy flavor modification to study the quark-gluon plasma produced in Pb-Pb collisions. The ALICE, ATLAS and CMS experiments are well-suited to measure bulk properties and select hard probes over a wide momentum range.
1) The document describes projects to obtain timing figures for orbit tracking filters using different satellite data window lengths.
2) Timing runs show the Gauss-Newton filter takes 0.484 seconds while the Kalman and Swerling filters take around 0.5 seconds for a 500,000 data point run.
3) Results are shown for the Molniya and GPS-10 satellites using window lengths of 350, 650, and 1250 seconds, demonstrating that doubling the window length improves accuracy by a factor of around 2.8.
Aerodynamics Part I of 3 describes aerodynamics of wings and bodies in subsonic flight.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
This document provides solutions to 24 problems in special relativity from an undergraduate physics textbook. It was created by Charles Asman, Adam Monahan and Malcolm McMillan at the University of British Columbia for their physics students. The problems cover various topics in special relativity including time dilation, length contraction, relativistic Doppler shift, and Lorentz transformations. Standard frames of reference and equations are defined. Detailed step-by-step solutions are provided for each problem.
The document discusses response spectra, which are plots of the maximum response of single-degree-of-freedom oscillators versus their natural period when subjected to a specific ground motion. Response spectra allow characterization of earthquake ground motions and are commonly used in earthquake-resistant design. The analysis procedure using response spectra involves obtaining the spectral acceleration from the response spectrum curve based on an oscillator's period and calculating the maximum displacement, base shear, and overturning moment. This procedure can be extended to multi-degree-of-freedom structures. An example is provided to demonstrate using a response spectrum to determine interstory drifts and story shears in a two-story building.
This document contains lecture notes on ultrasound physics and imaging. It covers fundamental acoustic topics like wave equations, diffraction theory, and the Doppler effect. It discusses how ultrasound waves propagate and interact with tissue, including nonlinear effects. Image formation concepts are introduced, like signal modeling, processing, and statistics. Specific lectures will cover acoustic interactions with tissue, Doppler ultrasound, and special topics. The document provides equations and diagrams to explain longitudinal and shear waves, radiation of ultrasound from apertures, the near field, and focused transducers.
This document summarizes a presentation on using the 4th order Runge-Kutta method to model the trajectory of projectiles. It discusses ordinary differential equations, introduces the RK4 method, and applies it to model the trajectory of unguided missiles accounting for gravity and drag forces, as well as guided self-propelled missiles where thrust is also considered. Input parameters for a simulation of a North Korean missile are provided and the optimal range is determined. Limitations of the model and references are also noted.
Please show the whole code... Im very confused. Matlab code for ea.pdfarshiartpalace
Please show the whole code... I\'m very confused. Matlab code for earth orbit Using a software
package such as MATLAB, create a plot of an Earth orbit with a = 31, 800 km and e = 0.6 for 0
lessthanorequalto theta lessthanorequalto 2 pi. The origin of the plot should be Earth. Perigee
should be in the direction of the positive x axis (therta = 0). On this plot, you should label (by
hand) the semimajor axis, semiminor axis, radius of perigee, radius of apogee, and the
semiparameter (semilatus rectum) of the ellipse.
Solution
---------------------------------------------------------------------------------------------------------------------
-------------------------------------------------
Elliptic orbit:
%periapsis
rp = 500e3 + getradius(\'earth\');
%eccentricity
ec = .7;
%semimajor axis
a = getsemimajoraxis(rp,ec);
%apoapsis
ra = 2*a-rp;
%true anomaly
theta = [0:.1:360];
orbit = ellipse(theta,a,ec,\'earth\');
figure(1), plot(orbit.x,orbit.y); axis equal
Re = getradius(\'earth\');
x = Re*cosd(theta);
y = Re*sind(theta);
hold on
plot(x,y,\'color\',[0 1 1])
hold off
figure(2), plot(theta,orbit.V/1000)
xlabel(\'\\theta\'), ylabel(\'V (km/s)\')
%current time
c = clock;
%reset hour and second
c(4) = 12; %hour
c(5) = 0; %minute
c(6) = orbit.tau; %seconds
%ellapsed time
datetime(c)
%returns 14-Feb-2015 21:34:54, so it\'s about 9.5 hours later still on
%Monday
2
ans =
22-Feb-2015 21:34:54
-----------Elliptic orbit
%periapsis
rp = 500e3 + getradius(\'earth\');
%eccentricity
ec = .7;
%semimajor axis
a = getsemimajoraxis(rp,ec);
%apoapsis
ra = 2*a-rp;
%true anomaly
theta = [0:.1:360];
orbit = ellipse(theta,a,ec,\'earth\');
figure(1), plot(orbit.x,orbit.y); axis equal
Re = getradius(\'earth\');
x = Re*cosd(theta);
y = Re*sind(theta);
hold on
plot(x,y,\'color\',[0 1 1])
hold off
figure(2), plot(theta,orbit.V/1000)
xlabel(\'\\theta\'), ylabel(\'V (km/s)\')
%current time
c = clock;
%reset hour and second
c(4) = 12; %hour
c(5) = 0; %minute
c(6) = orbit.tau; %seconds
%ellapsed time
datetime(c)
%returns 14-Feb-2015 21:34:54, so it\'s about 9.5 hours later still on
%Monday
2
ans =
22-Feb-2015 21:34:54
3
The circular orbits at perigee and apogee are
7.6167 and 3.1997 km/s, respectively.
-----Parabolic orbit:
Sphere of influence is actually a chapter 5 concept, but is the distance at which the gravitational
forces
between two objects is equal. Thus,
While an approximation, if we just estimate this as the point between the sun and the earth
where
these forces balance, then if the orbital radius of Earth is , then
, and thus solution for r_E in gives
r_soi = getsphereofinfluence(\'earth\',\'sun\');
display([\'Radius from center of earth to sphere of influence is approximately \' num2str(r_soi)]);
%perifocal distance
rp = 500e3 + getradius(\'earth\');
%eccentricty
ec = 1; %always so for a a parabola
theta = -180:1:180;
orbit = parabola(theta,rp,\'earth\');
figure(3), plot(orbit.x,orbit.y); axis equal
Re = getradius(\'earth\');
x = Re*cosd(theta);
y = Re*sind(theta);
.
1. The document discusses causality and relativity of simultaneity using spacetime diagrams. It explains that nothing can travel faster than the speed of light based on analyses of pole-barn paradox scenarios and Lorentz transformations.
2. Key concepts from special relativity are reviewed, including Lorentz transformations, 4-vectors like 4-displacement and 4-velocity, proper time, rest mass, 4-momentum, and conservation of 4-momentum. Collisions are also discussed in relation to conservation of 4-momentum.
3. Causality is maintained in special relativity because the order of events cannot be reversed between reference frames if they are causally connected within or on each other's light cones. Faster-
This document outlines the key aspects of using particle-based Monte Carlo simulations to solve the Boltzmann transport equation (BTE) for modeling semiconductor device transport. It describes how the BTE can be solved by decomposing carrier transport into free flight periods between scattering events. Random flight times are generated from the probability distribution of scattering rates. After each free flight, a scattering mechanism is chosen randomly based on its probability. New carrier momentum and energy are determined after each scattering event to model transport.
This document contains solutions to 5 questions about orbital mechanics. Question 1 calculates the centripetal and centrifugal accelerations, velocity, and orbital period of a satellite in a 1,400 km circular orbit. Question 2 does similar calculations for a 322 km circular orbit, finding the orbital angular velocity, period, and velocity. Question 3 calculates Doppler shifts for signals from this satellite received by observers in space and on the Earth's surface. Question 4 states Kepler's laws of planetary motion and uses the third law to find the orbital period of a satellite in an elliptical orbit with a 39,152 km apogee and 500 km perigee.
1) A student pointed out that a sailboat with a fan blowing into its sails could in fact move forward, because the air molecules bouncing off the sail would impart twice the momentum change experienced going through the fan, providing a net forward force.
2) A 2000-kg car traveling at 30 m/s collided perfectly inelastically with another 2000-kg car traveling at 10 m/s, sticking together with a final speed of 20 m/s. 20% of the initial kinetic energy was lost to heat and deformation.
3) A 16-g bullet fired into a 1.5-kg ballistic pendulum was calculated to have a speed of 45 m/s before impact by applying
The document discusses kinematics of particles, including rectilinear and curvilinear motion. It defines key concepts like displacement, velocity, and acceleration. It presents equations for calculating these values for rectilinear motion under different conditions of acceleration, such as constant acceleration, acceleration as a function of time, velocity, or displacement. Graphical interpretations are also described. An example problem is worked through to demonstrate finding velocity, acceleration, and displacement at different times for a particle moving in a straight line.
The document discusses several physics concepts:
1) Light interference from two slits and diffraction gratings, showing how the intensity of light varies with angle due to interference.
2) Seismic wave propagation through the Earth, describing how travel times vary with angle depending on the wave path through the mantle or core.
3) Vibrations of coupled harmonic oscillators arranged in linear chains, showing the normal mode solutions and frequency distributions that arise.
This document contains conceptual problems and their solutions related to motion in two and three dimensions. It discusses concepts such as displacement vs distance traveled, examples of motion with different acceleration and velocity vector directions, and solving problems involving velocity, acceleration, and displacement vectors. Sample problems include analyzing the motion of a dart thrown upward or falling downward, determining displacement vectors, and solving constant acceleration problems for objects moving in two dimensions.
How to learn Physics for JEE Main / MH-CET 2014Ednexa
This document contains two physics questions and their solutions. The first question asks about the force of friction on a 10kg block with a coefficient of friction of 0.6 when a 50N force is applied. The solution shows that the frictional force is 60N, but since the applied force is only 50N, the block will not move. The second question asks about the number of beats that would be heard from three tuning forks with frequencies of 400Hz, 401Hz, and 402Hz sounded together. The solution states that there would be two beats per second, or a total of 6 beats.
1) The document presents three theoretical physics problems involving spinning balls, charged particles in loops, and laser cooling of atoms.
2) Problem 1 considers a spinning ball falling and rebounding, calculating the rebound angle, horizontal distance traveled, and minimum spin rate. Problem 2 analyzes relativistic effects on charged particles in a moving loop in an electric field.
3) Problem 3 describes using lasers to cool atoms by resonant absorption. It calculates the laser frequency needed, velocity range absorbed, direction change upon emission, maximum velocity decrease, number of absorption events to slow to zero velocity, and distance traveled during cooling.
This document contains a summary of several physics concepts related to wave-particle duality and quantum physics. It includes 3 sample problems worked out in detail that demonstrate: 1) using the Compton scattering equation to estimate the Compton wavelength from experimental data, 2) relating the number of photons emitted by a laser to its power and photon energy, and 3) calculating the energy of the most energetic electron in uranium using the particle in a box model. The worked problems provide insight into applying relevant equations and show the conceptual and mathematical steps.
(1) The document provides conceptual problems and their solutions related to oscillations and simple harmonic motion. (2) It examines the kinetic and potential energy of an object undergoing simple harmonic motion with a given amplitude. (3) It compares the maximum speeds of two simple harmonic oscillators with identical amplitudes but different masses attached to identical springs.
The document discusses the Lorentz transformations, which replace the Galilean transformations of position and time. It derives the Lorentz transformations for position and time, which relate the coordinates of an event in one inertial reference frame to those in another frame moving at constant velocity. The inverse transformations are also derived. An example application of the transformations is provided.
This document summarizes solutions to three theoretical questions:
1) Describes how to use measurements of gravitational redshift to determine the mass and radius of a star.
2) Explains Snell's law and how it can be used to determine the path of light rays through a medium with a linear change in refractive index.
3) Analyzes the motion of a floating cylindrical buoy, determining equations for its vertical and rotational oscillations and relating the periods.
1. The document provides conceptual problems and solutions related to superposition and standing waves. It discusses topics like wave pulses traveling in opposite directions, fundamental frequencies of open and closed organ pipes, and using resonance frequencies to estimate air temperature.
2. It also covers problems involving interference of two waves with different phases and frequencies, and deriving an expression for the envelope of a superposed wave.
3. For one problem, it plots the total displacement of a superposed wave at t=0, and the envelope function at t=0, 5, and 10 seconds. From these plots, it estimates the speed of the envelope and compares it to the theoretical value obtained from the problem parameters.
The pearled solar eclipse of 1912.04.17 occurred 60 hours after the TITANIC disaster had cast its shadow upon this exciting event. The data collected during this most elusive eclipse are compared to those generated by Xavier JUBIER's 5MCSE, the most up-to date ergonomical solar eclipse simulation freeware, which allows the choice of the DeltaT parameter, as well as the exact GPS Coordinates of the observation site such as the balloon Globule at 900 meter over Rethondes.
This document outlines research on using aerodynamic drag to stabilize spacecraft attitude and target re-entry locations. Key points:
- A Drag De-Orbit Device (D3) is proposed that can be attached to CubeSats to control drag and expedite deorbiting. It has retractable tape spring booms to modulate drag area.
- Algorithms are presented for guidance trajectory generation, navigation with Kalman filtering, and guidance tracking. The algorithms compute drag profiles to guide a spacecraft along a trajectory to re-enter over a desired location.
- Hardware and simulation results show the D3 provides passive attitude stability and the algorithms can target re-entry points. Future work involves testing the D3
The document describes three models of photons with physical extent beyond the traditional point particle model: a KdV particle, a normal probability classical packet, and a sinc function quantum packet. The sinc function model is identified as most suitable, describing a photon peaked at its origin that converges to ±∞. In this model, the photon has a disk shape with radii ranging from 10-17m for gamma rays to unlimited sizes for long radio wavelengths. The photon is proposed to have internal magnetic fields and a possible rest mass upper limit of 2×10-69kg.
COLLEGE
PHYSICS LAB REPORT
STUDENTS NAME
ANALYSIS OF A BUBBLE CHAMBER PICTURE
SUPERVISED BY:
19/05/2020
1. Introduction
A bubble chamber is a vessel filled with a superheated transparent liquid (most often liquid hydrogen) used to detect electrically charged particles moving through it. It was invented in 1952 by Donald A. Glaser, for which he was awarded the 1960 Nobel Prize in Physics.
A convenient way to study the properties of the fundamental subatomic particles is through observation of their bubble trails, or tracks, in a bubble chamber. Using measurements made directly on a bubble chamber photograph, we can often identify the particles from their tracks and calculate their masses and other properties. In a typical experiment, a beam of a particular type of particle is sent from an accelerator into a bubble chamber, which is a large liquid-filled vessel. To simplify the analysis of the data, the liquid used is often hydrogen, the simplest element. The use of liquid hydrogen, while it simplifies the analysis, complicates the experiment itself, since hydrogen, a gas at room temperature, liquefies only when cooled to -246◦C. For charged particles to leave tracks in passing through the chamber, the liquid must be in a “super-heated” state, in which the slightest disturbance causes boiling to occur. In practice, this is accomplished by expanding the vapor above the liquid with a piston a few thousandths of a second before the particles enter the chamber.
2. Methods
2.1 Materials needed:
1. student worksheet per student
2. Ruler
3. Scissors
4. Glue stick
5. Pocket calculator
2.2 Procedures
2.2.1 Calculation of the X Particle’s Mass.
Make measurements on each of the photographs. In particular, for each of the circled events measure these four quantities:
· `Σ - The length of the Σ track,
· θ - the angle between the Σ− and π− track,
· s - the sagitta of the π− track,
· `π - The chord length of the π− track.
Your values for the event should be close to those given in the sample input. Run the program using each set of measurements, and tabulate the computed X0 mass from each event. Compute an average of the calculated masses and find the average deviation, expressing your result as Mx ±∆Mx.
Compare your final result with some known neutral particles listed below and identify the X0 particle based on this comparison.
Particlemass (in MeV/c2)
π0 135
K0 498
n 940
Λ0 1116
Σ0 1192
Ξ0 1315
2.2.2 Determination of the Angle θ.
The angle θ between the π− and Σ− momentum vectors can be determined by drawing tangents to the π− and Σ− tracks at the point of the Σ− decay.
We can then measure the angle between the tangents using a protractor. We can show.
This document summarizes information from a textbook chapter on elementary particles and the beginning of the universe. It provides examples of how to identify unknown particles in decay reactions using conservation laws of charge number, baryon number, strangeness, and spin. It also gives possible quark combinations for specific baryon particles. One example calculates the distance to a galaxy receding from Earth at 2.5% the speed of light using Hubble's law.
The document provides information about calculating binding energies of nuclei from their masses. It gives the calculations for the binding energy and binding energy per nucleon for 12C, 56Fe, and 238U. The thermal energy of neutrons at 25°C is also calculated, as is the half-life of neutrons based on the reduction of intensity of a monoenergetic neutron beam over distance. Rutherford's hypothesis that electrons could exist within the nucleus is shown to be invalid due to the high kinetic energies electrons would have based on the Heisenberg uncertainty principle.
This document contains conceptual problems and their solutions related to solids and condensed matter physics.
The key points summarized are:
1) When copper and brass samples are cooled from 300K to 4K, copper's resistivity decreases more because brass' resistivity at 4K is mainly due to impurities like zinc ions, while pure copper has very low residual resistance.
2) As temperature increases, copper's resistivity increases while silicon's decreases because silicon's number of charge carriers increases with temperature.
3) Calculations are shown to determine the free electron density, Fermi energy, and other properties of gold using given values and equations relating these concepts.
4) Resistivity and mean
1. The document discusses the rotational energy levels of a carbon monoxide (CO) molecule.
2. It is found that the characteristic rotational energy (E0r) of CO is 0.239 meV and an energy level diagram is drawn for rotational levels from l=0 to l=5.
3. Transitions between these levels that obey the selection rule Δl=-1 are indicated and the energies of the photons emitted during these transitions are calculated.
4. The wavelengths of the photons are then found, and it is determined that they fall within the microwave region of the electromagnetic spectrum.
The document discusses concepts related to the quantum mechanical model of the hydrogen atom. It provides answers to conceptual problems involving energy levels, quantum numbers, and properties of atomic orbitals. Key points include:
- As the principal quantum number n increases, the spacing between adjacent energy levels decreases.
- For n=4, the orbital quantum number l can take on values from 0 to 3.
- In sodium, the 3s state is at a lower energy than the 3p state due to penetration of the 3s orbital closer to the nucleus. In hydrogen, the 3s and 3p states have similar energies.
- The Ritz combination principle, where 1/λ1 + 1/λ2
This document derives the energy of the first excited state of the harmonic oscillator using the Schrödinger equation. It first shows that the wave function for the first excited state is a1xe−ax2, where a is a constant. It then substitutes this wave function and its derivatives into the Schrödinger equation for the harmonic oscillator to obtain an expression for the energy E1 in terms of the angular frequency ω. Solving this expression yields the result that the energy of the first excited state is E1 = (3/2)ħω.
- Newton's rings are an interference pattern produced when a plano-convex lens rests on a flat glass plate, creating an air gap of variable thickness.
- The condition for constructive interference and bright rings is that the thickness equals an integer multiple of half the wavelength plus a quarter wavelength.
- For small thicknesses compared to the lens's radius of curvature, the radius of each bright ring is proportional to the square of the thickness.
- Given a 10m radius lens illuminated with 590nm light, over 40 bright rings would be seen within the 4cm diameter, and the sixth ring would have a diameter of around 0.5cm.
- Replacing the air with water of a higher index would make
This document contains conceptual problems and their solutions related to optical images formed by mirrors and lenses. For concave mirrors, it discusses that the virtual image size depends on the object distance, and real images are possible. Convex mirrors never form real images. A concave mirror can form enlarged real images if the object is between the center of curvature and focal point. Plane mirrors form virtual images, and the eye location range to see the image is discussed. Spherical mirrors equations relate image and object distances. Refraction through a fish bowl or glass rod immersed in water is analyzed. A double concave lens problem applies lens equations to find the focal length, image location and size, and determines if the image is real/virtual and
The document contains conceptual problems and their solutions related to properties of light.
1. A ray of light reflects from a plane mirror at an angle of 70° between the incoming and reflected rays. The angle of reflection is 35°.
2. A lifeguard hears a swimmer calling for help. Taking the least time path, the lifeguard chooses to run on land then swim through point D to reach the swimmer.
3. Blue light appears blue underwater because the color molecules in the eye respond to the frequency of light, not the wavelength, which changes with the medium's index of refraction.
This document contains conceptual problems and their solutions related to alternating current circuits. It discusses key concepts like how changing the frequency of an AC circuit affects different circuit components. For example, it explains that while the inductance of an inductor remains the same with changing frequency, the inductive reactance is frequency dependent. It also provides examples of calculating current, power, reactance, and oscillation periods in various AC circuits containing resistors, capacitors and inductors. The problems cover a range of concepts involving driven and undriven circuits.
The document contains a multi-part conceptual physics problem about magnetic induction.
Part 1 asks about orienting a sheet of paper at the magnetic equator to maximize or minimize magnetic flux through it. Part 2 shows that the units T⋅m2/s are equivalent to volts. Part 3 asks about the direction of induced current in a conducting loop based on the direction and changing magnitude of an applied magnetic field.
1) Gauss's law for magnetism states that the magnetic flux through a closed surface is always zero, since there is no magnetic monopole. Gauss's law for electricity relates the electric flux through a closed surface to the net electric charge enclosed.
2) Applying the right-hand rule, the direction of the current in a solenoid that produces a magnetic field pointing away from you is clockwise.
3) Of the gases listed, H2, CO2, and N2 are diamagnetic with magnetic susceptibility χm < 0, while O2 is paramagnetic with χm > 0.
1. A particle moving perpendicular to a magnetic field will follow a circular path. The radius of the path is determined by the particle's mass, charge, speed, and the magnetic field strength.
2. A velocity selector uses uniform, perpendicular electric and magnetic fields. Particles pass through undeflected if their speed equals the ratio of the field strengths.
3. A mass spectrometer accelerates ions and uses a magnetic field to cause circular orbits. Heavier ions have smaller orbit radii allowing separation based on mass.
1) The document discusses various concepts related to electric current and direct current circuits, including resistance, resistivity, Ohm's law, temperature dependence of resistance, and drift velocity of electrons.
2) It provides examples and conceptual problems involving calculating resistance, current, power, drift velocity, and other quantities using relationships like Ohm's law, relationships between resistance and resistivity, and how resistance changes with temperature.
3) Sample problems include calculating the resistance of wires with different lengths and gauges, the drift speed of electrons in copper wires carrying different currents, the potential difference required across a wire to produce a given current, and the temperature at which a copper wire's resistance would be 10% greater than at 20
(1) The document discusses capacitance concepts including the capacitance of parallel plate, cylindrical, and spherical capacitors. (2) It provides equations to calculate the capacitance, electric field, and energy stored in capacitors. (3) Key equations include expressions for capacitance as a function of plate area and separation, and energy stored as proportional to capacitance times the square of the potential difference.
This document contains conceptual problems and their solutions related to electric fields and Gauss's law.
Problem 29 asks about an electric field given by a formula and calculates (a) the electric flux through each end of a cylinder in that field, (b) the flux through the curved surface, and (c) the total flux through the closed cylindrical surface. It then (d) uses Gauss's law to find the net charge inside the cylinder.
Problem 33 gives the electric flux out of one side of an imaginary cube and asks the reader to use Gauss's law to determine the amount of charge at the center of the cube.
1) Two point charges are located on the y-axis separated by a distance d. The electric field is sketched in the neighborhood and at distances much greater than d. At greater distances, the system looks like a single charge equal to the sum of the charges.
2) The document discusses dipole moments of two molecules and determining the electric field direction at numbered locations based on the orientation of the dipole moments.
3) Information is provided about the total charge of protons in 1 kg of carbon, including using proportions and other relationships to calculate the number of atoms and coulombs of charge.
1. Placing a full glass bottle of water in the freezer would cause it to break because water expands as it freezes and the sealed bottle provides no room for expansion.
2. The phase diagram shows that at higher altitudes, the boiling point of water decreases and the melting point increases due to lower atmospheric pressure. This could require longer cooking times in mountains.
3. If two cylinders made of materials A and B conduct heat at the same rate when subjected to the same temperature difference, and the diameter of A is twice the diameter of B, then the thermal conductivity of A is one fourth that of B.
1) The document discusses the processes involved in a Carnot cycle for an ideal gas, including isothermal expansion and compression and adiabatic processes.
2) It examines the efficiencies of Carnot engines and refrigerators, noting that engines are more efficient when the temperature difference is large, while refrigerators are more efficient when the temperature difference is small.
3) It then shows how assuming the heat engine statement of the second law is false would allow using a refrigerator to violate the refrigerator statement of the second law by creating a perpetual motion machine.
DEEP LEARNING FOR SMART GRID INTRUSION DETECTION: A HYBRID CNN-LSTM-BASED MODELgerogepatton
As digital technology becomes more deeply embedded in power systems, protecting the communication
networks of Smart Grids (SG) has emerged as a critical concern. Distributed Network Protocol 3 (DNP3)
represents a multi-tiered application layer protocol extensively utilized in Supervisory Control and Data
Acquisition (SCADA)-based smart grids to facilitate real-time data gathering and control functionalities.
Robust Intrusion Detection Systems (IDS) are necessary for early threat detection and mitigation because
of the interconnection of these networks, which makes them vulnerable to a variety of cyberattacks. To
solve this issue, this paper develops a hybrid Deep Learning (DL) model specifically designed for intrusion
detection in smart grids. The proposed approach is a combination of the Convolutional Neural Network
(CNN) and the Long-Short-Term Memory algorithms (LSTM). We employed a recent intrusion detection
dataset (DNP3), which focuses on unauthorized commands and Denial of Service (DoS) cyberattacks, to
train and test our model. The results of our experiments show that our CNN-LSTM method is much better
at finding smart grid intrusions than other deep learning algorithms used for classification. In addition,
our proposed approach improves accuracy, precision, recall, and F1 score, achieving a high detection
accuracy rate of 99.50%.
KuberTENes Birthday Bash Guadalajara - K8sGPT first impressionsVictor Morales
K8sGPT is a tool that analyzes and diagnoses Kubernetes clusters. This presentation was used to share the requirements and dependencies to deploy K8sGPT in a local environment.
CHINA’S GEO-ECONOMIC OUTREACH IN CENTRAL ASIAN COUNTRIES AND FUTURE PROSPECTjpsjournal1
The rivalry between prominent international actors for dominance over Central Asia's hydrocarbon
reserves and the ancient silk trade route, along with China's diplomatic endeavours in the area, has been
referred to as the "New Great Game." This research centres on the power struggle, considering
geopolitical, geostrategic, and geoeconomic variables. Topics including trade, political hegemony, oil
politics, and conventional and nontraditional security are all explored and explained by the researcher.
Using Mackinder's Heartland, Spykman Rimland, and Hegemonic Stability theories, examines China's role
in Central Asia. This study adheres to the empirical epistemological method and has taken care of
objectivity. This study analyze primary and secondary research documents critically to elaborate role of
china’s geo economic outreach in central Asian countries and its future prospect. China is thriving in trade,
pipeline politics, and winning states, according to this study, thanks to important instruments like the
Shanghai Cooperation Organisation and the Belt and Road Economic Initiative. According to this study,
China is seeing significant success in commerce, pipeline politics, and gaining influence on other
governments. This success may be attributed to the effective utilisation of key tools such as the Shanghai
Cooperation Organisation and the Belt and Road Economic Initiative.
Literature Review Basics and Understanding Reference Management.pptxDr Ramhari Poudyal
Three-day training on academic research focuses on analytical tools at United Technical College, supported by the University Grant Commission, Nepal. 24-26 May 2024
Understanding Inductive Bias in Machine LearningSUTEJAS
This presentation explores the concept of inductive bias in machine learning. It explains how algorithms come with built-in assumptions and preferences that guide the learning process. You'll learn about the different types of inductive bias and how they can impact the performance and generalizability of machine learning models.
The presentation also covers the positive and negative aspects of inductive bias, along with strategies for mitigating potential drawbacks. We'll explore examples of how bias manifests in algorithms like neural networks and decision trees.
By understanding inductive bias, you can gain valuable insights into how machine learning models work and make informed decisions when building and deploying them.
Optimizing Gradle Builds - Gradle DPE Tour Berlin 2024Sinan KOZAK
Sinan from the Delivery Hero mobile infrastructure engineering team shares a deep dive into performance acceleration with Gradle build cache optimizations. Sinan shares their journey into solving complex build-cache problems that affect Gradle builds. By understanding the challenges and solutions found in our journey, we aim to demonstrate the possibilities for faster builds. The case study reveals how overlapping outputs and cache misconfigurations led to significant increases in build times, especially as the project scaled up with numerous modules using Paparazzi tests. The journey from diagnosing to defeating cache issues offers invaluable lessons on maintaining cache integrity without sacrificing functionality.
Optimizing Gradle Builds - Gradle DPE Tour Berlin 2024
Ch39 ssm
1. Chapter 39
Relativity
Conceptual Problems
1 • The approximate total energy of a particle of mass m moving at speed
u << c is (a)mc2
+ 1
2
mu2
, (b) 1
2
mu2
, (c) cmu, (d) mc2
, (e) 1
2
cmu .
Determine the Concept The total relativistic energy E of a particle is defined to
be the sum of its kinetic and rest energies.
The sum of the kinetic and rest
energies of a particle is given by:
22
2
12
mcmumcKE +=+=
and )(a is correct.
Estimation and Approximation
7 •• The most distant galaxies that can be seen by the Hubble telescope are
moving away from us and have a redshift parameter of about z = 5. (The redshift
parameter z is defined as (f – f’)/f’, where f is the frequency measured in the rest
frame of the emitter, and f’ is the frequency measured in the rest frame of the
receiver.) (a) What is the speed of these galaxies relative to us (expressed as a
fraction of the speed of light)? (b) Hubble’s law states that the recession speed is
given by the expression v = Hx, where v is the speed of recession, x is the
distance, and H, the Hubble constant, is equal to 75 km/s/Mpc , where
1 pc = 3.26 c⋅y. (The abbreviation for parsec is pc.) Estimate the distance of such
a galaxy from us using the information given.
Picture the Problem (a) We can use the definition of the redshift parameter and
the relativistic Doppler shift equation to show that, for light that is Doppler-
shifted with respect to an observer, ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+
−
=
1
1
2
2
u
u
cv , where u = z + 1, and to find the
ratio of v to c. In Part (b) we can solve Hubble’s law for x and substitute our result
from Part (a) to estimate the distance to the galaxy.
(a) The red-shift parameter is defined
to be: f'
'ff
z
−
= 0
The relativistic Doppler shift for
recession is given by:
cv
cv
f'f
+
−
=
1
1
0
37
2. Chapter 3938
Substitute for f ′ and simplify to
obtain:
1
1
1
1
1
1
11
0
00
−
−
+
=
+
−
+
−
=
cv
cv
cv
cv
f
cv
cv
ff
z
Letting u = z + 1 and simplifying
yields:
cv
cv
zu
−
+
=+=
1
1
1 ⇒
1
1
2
2
+
−
=
u
u
c
v
Substitute for u to express v/c as a
function of z:
( )
( ) 11
11
2
2
++
−+
=
z
z
c
v
Substituting the numerical value of z
and evaluating v/c gives:
( )
( )
946.0
115
115
2
2
=
++
−+
=
c
v
(b) Solving Hubble’s law for x
yields: H
v
x =
Substitute numerical values and
evaluate x:
yG3.12
Mpc
y103.26
Mpc
km/s
75
946.0946.0 6
⋅=
⋅×
×==
c
cc
H
c
x
Time Dilation and Length Contraction
13 •• Unobtainium (Un) is an unstable particle that decays into normalium
(Nr) and standardium (St) particles. (a) An accelerator produces a beam of Un
that travels to a detector located 100 m away from the accelerator. The particles
travel with a velocity of v = 0.866c. How long do the particles take (in the
laboratory frame) to get to the detector? (b) By the time the particles get to the
detector, half of the particles have decayed. What is the half-life of Un? (Note:
half-life as it would be measured in a frame moving with the particles) (c) A new
detector is going to be used, which is located 1000 m away from the accelerator.
How fast should the particles be moving if half of the particles are to make it to
the new detector?
3. Relativity 39
Picture the Problem The time required for the particles to reach the detector, as
measured in the laboratory frame of reference is the ratio of the distance they
must travel to their speed. The half life of the particles is the trip time as measured
in a frame traveling with the particles. We can find the speed at which the
particles must move if they are to reach the more distant detector by equating their
half life to the ratio of the distance to the detector in the particle’s frame of
reference to their speed.
(a) The time required to reach the
detector is the ratio of the distance to
the detector and the speed with
which the particles are traveling:
c
x
v
x
t
866.0
Δ
=
Δ
=Δ
Substitute numerical values and
evaluate Δt: ( )
s385.0
m/s10998.2866.0
m100
Δ 8
μ=
×
=t
(b) The half life is the trip time as
measured in a frame traveling with
the particles:
2
1 ⎟
⎠
⎞
⎜
⎝
⎛
−Δ=
Δ
=Δ
c
v
t
t
t'
γ
Substitute numerical values and
evaluate Δt′: ( )
s193.0
866.0
1s385.0
2
μ
μ
=
⎟
⎠
⎞
⎜
⎝
⎛
−=Δ
c
c
t'
(c) In order for half the particles to
reach the detector:
v
c
v
x'
v
x'
t'
2
1 ⎟
⎠
⎞
⎜
⎝
⎛
−Δ
=
Δ
=Δ
γ
where Δx′ is the distance to the new
detector.
Rewrite this expression to obtain:
t'
x'
c
v
v
Δ
Δ
=
⎟
⎠
⎞
⎜
⎝
⎛
−
2
1
Squaring both sides of the equation
yields:
2
2
2
1
⎟
⎠
⎞
⎜
⎝
⎛
Δ
Δ
=
⎟
⎠
⎞
⎜
⎝
⎛
−
t'
x'
c
v
v
4. Chapter 3940
Substitute numerical values for Δx′
and Δt′ and simplify to obtain: ( )2
2
2
2
3.17
s193.0
m1000
1
c
c
v
v
=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
⎟
⎠
⎞
⎜
⎝
⎛
−
μ
Divide both sides of the equation by
c2
to obtain: ( )2
2
2
2
3.17
1
=
⎟
⎠
⎞
⎜
⎝
⎛
−
c
v
c
v
Solving this equation for v2
/c2
gives: ( )
( )
9967.0
3.171
3.17
2
2
2
2
=
+
=
c
v
Finally, solving for v yields: cv 998.0=
The Lorentz Transformation, Clock Synchronization, and
Simultaneity
17 •• A spaceship of proper length Lp = 400 m moves past a transmitting
station at a speed of 0.760c. (The transmitting station broadcasts signals that
travel at the speed of light.) A clock is attached to the nose of the spaceship and a
second clock is attached to the transmitting station. The instant that the nose of
the spaceship passes the transmitter, the clock attached to the transmitter and the
clock attached to the nose of the spaceship are set equal to zero. The instant that
the tail of the spaceship passes the transmitter a signal is sent by the transmitter
that is subsequently detected by a receiver in the nose of the spaceship. (a) When,
according to the clock attached to the nose of spaceship, is the signal sent?
(b) When, according to the clocks attached to the nose of spaceship, is the signal
received? (c) When, according to the clock attached to the transmitter, is the
signal received by the spaceship? (d) According to an observer that works at the
transmitting station, how far from the transmitter is the nose of the spaceship
when the signal is received?
Picture the Problem Let S be the reference frame of the spaceship and S′ be that
of Earth (transmitter station). Let event A be the emission of the light pulse and
event B the reception of the light pulse at the nose of the spaceship. In (a) and (c)
we can use the classical distance, rate, and time relationship and in (b) and (d) we
can apply the inverse Lorentz transformations.
(a) In both S and S′ the pulse travels
at the speed c. Thus:
s76.1
0.760c
m400p
A μ===
v
L
t
5. Relativity 41
(b) The inverse time transformation
is:
⎟
⎠
⎞
⎜
⎝
⎛
−= 2B
c
vx
t't γ
where
( )
54.1
760.0
1
1
1
1
2
2
2
2
=
−
=
−
=
c
c
c
v
γ
Substitute numerical values and
evaluate :'tB
( ) ( )( )
( ) ( )(
( )
)
s76.4
m/s10998.2
m400760.0
s09.354.1
m400760.0
s09.354.1
28
2B
μ
μ
μ
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
×
−
−=
⎟
⎠
⎞
⎜
⎝
⎛ −
−=
c
c
't
(c) The elapsed time, according to
the clock on the ship is:
A
shipoflength
traveltopulseB ttt +=
Find the time of travel of the pulse to
the nose of the ship:
s33.1
m/s102.998
m400
8
shipoflength
traveltopulse
μ=
×
=t
Substitute numerical values and
evaluate tB:
s09.3s76.1s33.1B μμμ =+=t
(d) The inverse transformation for x
is:
( )vtxx' −= γ
Substitute numerical values and evaluate x′:
( ) ( )( )( )[ ] km70.1s1009.3m/s10998.2760.0m40054.1 68
=××−−= −
x'
The Relativistic Doppler Shift
27 •• A clock is placed in a satellite that orbits Earth with an orbital period
of 90 min. By what time interval will this clock differ from an identical clock on
Earth after 1.0 y? (Assume that special relativity applies and neglect general
relativity.)
Picture the Problem Due to its motion, the orbiting clock will run more slowly
than the Earth-bound clock. We can use Kepler’s third law to find the radius of
6. Chapter 3942
the satellite’s orbit in terms of its period, the definition of speed to find the orbital
speed of the satellite from the radius of its orbit, and the time dilation equation to
find the difference δ in the readings of the two clocks.
Express the time δ lost by the clock:
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−Δ=
Δ
−Δ=Δ−Δ=
γγ
δ
1
1p t
t
ttt
Because v << c, we can use Part (b)
of Problem 10: 2
2
2
1
1
1
c
v
−≈
γ
Substitute to obtain:
t
c
v
c
v
t Δ=⎥
⎦
⎤
⎢
⎣
⎡
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−−Δ= 2
2
2
2
2
1
2
1
11δ (1)
Express the square of the speed of
the satellite in its orbit: 2
222
2 42
T
r
T
r
v
ππ
=⎟
⎠
⎞
⎜
⎝
⎛
= (2)
where T is its period and r is the radius
of its (assumed) circular orbit.
Use Kepler’s third law to relate the
period of the satellite to the radius of
its orbit about Earth:
3
2
E
2
3
E
2
2 44
r
gR
r
GM
T
ππ
== ⇒ 3
2
22
E
4π
TgR
r =
Substitute numerical values and evaluate r:
( )( ) ( ) m1065.6
4
s/min60min90km6370m/s81.9 63
2
222
×=
×
=
π
r
Substitute numerical values in
equation (2) and evaluate v2
:
( )
( )
227
2
262
2
s/m1099.5
s/min60min90
m1065.64
×=
×
×
=
π
v
Finally, substitute for v2
in equation (1) and evaluate δ:
( )( )
( )
ms11
m/s10998.2
Ms/y31.56y0.1s/m1099.5
2
1
28
227
=
×
××
=δ
31 •• A particle moves with speed 0.800c in the + ′′x direction along the ′′x
axis of frame , which moves with the same speed and in the same direction
along the x′ axis relative to frame S′. Frame S′ moves with the same speed and in
′′S
7. Relativity 43
the same direction along the x axis relative to frame S. (a) Find the speed of the
particle relative to frame S′. (b) Find the speed of the particle relative to frame S.
Picture the Problem We can apply the inverse velocity transformation equation
to express the speed of the particle relative to both frames of reference.
(a) Express in terms of'ux :''ux
2
1
c
''vu
v''u
'u
x
x
x
+
+
=
where v of S ′, relative to S″, is 0.800c.
Substitute numerical values and
evaluate :'ux ( )
c
c
c
c
cc
'ux
976.0
64.1
60.1
800.0
1
800.0800.0
2
2
=
=
+
+
=
(b) Express ux in terms of :'ux
2
1
c
'vu
v'u
u
x
x
x
+
+
= where v, the speed of S,
relative to S ′, is 0.800c.
( )( )
c
c
c
cc
cc
ux
997.0
781.1
776.1
976.0800.0
1
800.0976.0
2
=
=
+
+
=Substitute numerical values and
evaluate ux:
Relativistic Momentum and Relativistic Energy
37 •• In reference frame S’, two protons, each moving at 0.500c, approach
each other head-on. (a) Calculate the total kinetic energy of the two protons in
frame S′. (b) Calculate the total kinetic energy of the protons as seen in reference
frame S, which is moving with speed 0.500c relative to S′ so that one of the
protons is at rest.
Picture the Problem The total kinetic energy of the two protons in Part (a) is the
sum of their kinetic energies and is given by ( ) 012 EK −= γ . Part (b) differs from
Part (a) in that we need to find the speed of the moving proton relative to frame S.
8. Chapter 3944
(a) The total kinetic energy of the
protons in frame S′ is given by:
( ) 012 EK −= γ
Substitute for γ and E0 and evaluate
K:
( )
( )
MeV290
MeV28.9381
500.0
1
1
2
2
2
=
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎜
⎝
⎛
−
−
=
c
c
K
(b) The kinetic energy of the moving
proton in frame S is given by:
( ) 01 EK −= γ (1)
where
2
1
1
c
uv
−
=γ
Express the speed u of the proton
in frame S:
2
1
c
'vu
v'u
u
x
x
+
+
=
( )( ) c
c
cc
cc
u 800.0
500.0500.0
1
500.0500.0
2
=
+
+
=
Substitute numerical values and
evaluate u:
Evaluate γ:
( )( )
67.1
800.0800.0
1
1
2
=
−
=
c
cc
γ
( )( )
MeV629
MeV28.938167.1
=
−=KSubstitute numerical values in
equation (1) and evaluate K:
General Problems
45 •• Frames S and S′ are moving relative to each other along the x and x′
axes (which superpose). Observers at rest in the two frames set their clocks to
t = 0 when the two origins coincide. In frame S, event 1 occurs at x1 = 1.0 c⋅y and
t1 = 1.00 y and event 2 occurs at x2 = 2.0 c⋅y and t2 = 0.50 y. These events occur
simultaneously in frame S′. (a) Find the magnitude and direction of the velocity of
S′ relative to S. (b) At what time do both these events occur as measured in S′?
9. Relativity 45
Picture the Problem We can use Equation 39-12, the inverse time transformation
equation, to relate the elapsed times and separations of the events in the two
systems to the velocity of S′ relative to S. We can use this same relationship in
Part (b) to find the time at which these events occur as measured in S′.
(a) Use Equation 39-12 to obtain:
( ) ( )
⎥⎦
⎤
⎢⎣
⎡
Δ−Δ=
⎥⎦
⎤
⎢⎣
⎡
−−−=−=Δ
x
c
v
t
xx
c
v
tt't'tt'
2
1221212
γ
γ
Because the events occur
simultaneously in frame S′,
Δt′ = 0 and:
x
c
v
t Δ−Δ= 2
0 ⇒
x
tc
v
Δ
Δ
=
2
Substitute for Δt and Δx and evaluate
v:
( ) c
cc
c
v 50.0
y1.00y0.2
y1.00y50.02
−=
⋅−⋅
−
=
Because :y50.012 −=−= tttΔ direction.in themoves' xS −
(b) Use the inverse time
transformation to obtain:
2
2
2
2
2
2
2
22
1
c
v
c
vx
t
c
vx
t't
−
−
=⎟
⎠
⎞
⎜
⎝
⎛
−= γ
( )( )
( )
y7.1
50.0
1
y0.250.0
y50.0
2
2
2
12
=
−
−
⋅−
−
==
c
c
c
cc
't't
Substitute numerical values and
evaluate t2′ and t1′:
49 •• Using a simple thought experiment, Einstein showed that there is mass
associated with electromagnetic radiation. Consider a box of length L and mass M
resting on a frictionless surface. Attached to the left wall of the box is a light
source that emits a directed pulse of radiation of energy E, which is completely
absorbed at the right wall of the box. According to classical electromagnetic
theory, this radiation carries momentum of magnitude p = E/c (Equation 32-13).
The box recoils when the pulse is emitted by the light source. (a) Find the recoil
velocity of the box so that momentum is conserved when the light is emitted.
(Because p is small and M is large, you may use classical mechanics.) (b) When
the light is absorbed at the right wall of the box the box stops, so the total
10. Chapter 3946
momentum of the system remains zero. If we neglect the very small velocity of
the box, the time it takes for the radiation to travel across the box is Δt = L/c. Find
the distance moved by the box in this time. (c) Show that if the center of mass of
the system is to remain at the same place, the radiation must carry mass m = E/c2
.
Picture the Problem We can use conservation of energy to express the recoil
velocity of the box and the relationship between distance, speed, and time to find
the distance traveled by the box in time Δt = L/c. Equating the initial and final
locations of the center of mass will allow us to show that the radiation must carry
mass m = E/c2
.
(a) Apply conservation of momentum
to obtain:
0i ==+ pMv
c
E
⇒
Mc
E
v −=
c
vL
tvd =Δ=
(b) The distance traveled by the
box in time Δt = L/c is:
Substitute for v from (a) to obtain:
2
Mc
LE
Mc
E
c
L
d −=⎟
⎠
⎞
⎜
⎝
⎛
−=
mM
mL
x
+
−
= 2
1
CM
(c) Let x = 0 be at the center of the
box and let the mass of the photon be
m. Then initially the center of mass
is at:
When the photon is absorbed at the
other end of the box, the center of
mass is at: mM
Mc
EL
Lm
Mc
MEL
x
+
⎥
⎦
⎤
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
−+
−
=
22
1
2
CM
Because no external forces act on the
system, these expressions for xCM
must be equal: mM
Mc
EL
Lm
Mc
MEL
mM
mL
+
⎥
⎦
⎤
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
−+
−
=
+
− 22
1
2
2
1
Solving for m yields:
⎟
⎠
⎞
⎜
⎝
⎛
−
=
2
2
1
Mc
E
c
E
m
Because Mc2
is of the order of 1016
J
and E = hf is of the order of 1 J for
reasonable values of f, E/Mc2
<< 1
and:
2
c
E
m =
11. Relativity 47
51 ••• When a moving particle that has a kinetic energy greater than the
threshold kinetic energy Kth strikes a stationary target particle, one or more
particles may be created in the inelastic collision. Show that the threshold kinetic
energy of the moving particle is given by
Kth =
Σmin + Σmfin( ) Σmfin + Σmin( )c2
2mtarget
Here Σmin is the sum of the masses of the particles prior to the collision, Σmfin is
the sum of the masses of the particles following the collision, and mtarget is the
mass of the target particle. Use this expression to determine the threshold kinetic
energy of protons incident on a stationary proton target for the production of a
proton–antiproton pair; compare your result with the result of Problem 38.
Picture the Problem Let mi denote the mass of the incident (projectile) particle.
Then ∑min = mi + mtarget and we can use this expression to determine the threshold
kinetic energy of protons incident on a stationary proton target for the production
of a proton–antiproton pair.
Consider the situation in the center
of mass reference frame. At
threshold we have:
∑=− 2
fin
222
cmcpE
Note that this is a relativistically
invariant expression.
t,0ttarget EEE ==In the laboratory frame, the target is
at rest so:
We can, therefore, write: ( ) ( )22
fin
22
i
2
t,0i ∑=−+ cmcpEE
For the incident particle: 2
i,0
22
i
2
i EcpE =−
and
thi,0i KEE +=
where is the threshold kinetic
energy of the incident particle in the
laboratory frame.
thK
Express in terms of the rest
energies:
thK ( ) ( )22
fint,0th
2
i.0t,0 2 ∑=++ cmEKEE
where
∑=+ 2
fini.0t,0 cmEE
and
2
targett,0 cmE =
12. Chapter 3948
( ) ( 22
fin
2
targetth
22
fin 2 ∑∑ =+ cmcmKcm )Substitute to obtain:
Solving for gives:thK
( )( )
target
2
infinfinin
th
2m
cmmmm
K
∑ ∑∑ ∑ −+
=
For the creation of a proton -
antiproton pair in a proton - proton
collision:
∑ = pin 2mm , ∑ = pfin 4mm and
ptarget mm =
Substituting for the sums and
simplifying yields:
( )( )
( )( ) 2
p
p
2
pp
p
2
pppp
th
6
2
26
2
2442
cm
m
cmm
m
cmmmm
K
==
−+
=
in agreement with Problem 38.
55 ••• For the special case of a particle moving with speed u along the y axis
in frame S, show that its momentum and energy in frame S′ are related to its
momentum and energy in S by the transformation equations
⎟
⎠
⎞
⎜
⎝
⎛
−= 2
'
c
vE
pp xx γ , , , andyy pp ='
zz pp ='
⎟
⎠
⎞
⎜
⎝
⎛
−=
c
vp
c
E
c
E x
γ
'
.
Compare these equations with the Lorentz transformation equations for x′, y′, z′,
and t′. Notice that that the quantities px, py, pz, and E/c transform in the same way
as do x, y, z, and ct.
Picture the Problem We can use the expressions for p
r
and E in S together with
the relations we wish to verify and the inverse velocity transformation equations
to establish the condition ( ) ( ) ( ) 2
2
22222
γ
u
v'u'u'uu' zyx +=++= and then use this
result to verify the given expressions for px′, py′, pz′ and E′/c.
In any inertial frame the momentum
and energy are given by:
2
2
1
c
u
m
−
=
u
p
r
r
and
2
2
2
1
c
u
mc
E
−
=
where u
r
is the velocity of the particle
and u is its speed.
13. Relativity 49
The components of p
r
in S are:
2
2
1
c
u
mu
p x
x
−
= ,
2
2
1
c
u
mu
p
y
y
−
= , and
2
2
1
c
u
mu
p z
z
−
=
Because ux = uz = 0 and uy = u:
0== zx pp and
2
2
1
c
u
mu
py
−
=
Substituting zeros for px and pz in
the relations we are trying to
show yields:
22
0
c
vE
c
vE
'px γγ −=⎟
⎠
⎞
⎜
⎝
⎛
−= , ,yy p'p =
0='pz , and
c
E
c
E
c
E'
γγ =⎟
⎠
⎞
⎜
⎝
⎛
−= 0
In S′ the momentum components
are:
2
2
1
c
u'
'mu
'p x
x
−
= ,
2
2
1
c
u'
'mu
'p
y
y
−
= , and
2
2
1
c
u'
'mu
'p z
z
−
=
The inverse velocity transformations
are:
2
1
c
vu
vu
'u
x
x
x
−
−
= ,
2
1
c
vu
u
'u
y
y
y
−
= , and
2
1
c
vu
u
'u
z
z
z
−
=
Substitute ux = uz = 0 and uy = u to
obtain:
v'ux −= , , andu'uy γ= 0='uz
Thus: ( ) ( ) ( )
2
2
2
2222
γ
u
v
'u'u'uu' zyx
+=
++=
14. Chapter 3950
First we verify that pz′ = pz = 0: ( ) 0
1
0
2
2
==
−
= zz p
c
u'
m
'p
Next we verify that py′ = py:
y
y
y
y
p
c
v
c
u
c
v
c
v
c
u
c
v
p
c
v
c
u
c
v
c
v
c
u
c
u
mu
c
u
c
v
c
u
c
u
mu
c
u
c
v
mu
c
u'
'mu
'p
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−−−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−−−
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−−−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
−
=
−−
−
−
=
−−
=
−
=
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
22
2
2
2
2
2
2
2
22
2
2
2
2
2
11
11
11
11
1
1
1
111
γ
γ
γ
γ
Next, we verify that ⎟
⎠
⎞
⎜
⎝
⎛
−= 2
c
vE
p'p xx γ :
E
c
v
c
v
c
u
c
v
c
v
c
u
c
v
E
c
v
c
v
c
u
c
v
c
v
c
u
E
c
v
c
u
c
v
c
u
c
u
mc
c
v
c
u
c
v
mv
c
u'
'mu
'p x
x
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
22
2
2
2
2
2
1
2
2
2
2
22
2
2
2
2
2
11
11
11
11
1
1
111
γ
γγ
γ
γ
γ
γ
γ
−=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−−−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−−−
−=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−−−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
−=
−−
−
−
−=
−−
−
=
−
=
−
15. Relativity 51
Finally, we verify that :or, EE'
c
E
c
vp
c
E
c
E' x
γγγ ==⎟
⎠
⎞
⎜
⎝
⎛
−=
E
c
v
c
u
c
v
c
v
c
u
c
v
E
c
v
c
u
c
v
c
v
c
u
E
c
u
c
v
c
u
E
c
u'
c
u
c
u
mc
c
u'
mc
E'
γ
γγ
γ
γ
γ
γ
γ
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−−−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−−−
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−−−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
=
−−
−
=
−
−
−
=
−
=
−−
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
22
2
2
2
2
2
1
2
2
2
2
1
2
2
2
2
2
2
11
11
11
11
1
1
1
1
11
The x, y, z, and t transformation
equations are:
( )vtxx' −= γ , yy' = , zz' =
and
⎟
⎠
⎞
⎜
⎝
⎛
−= 2
c
vx
tt' γ
The x, y, z, and ct transformation
equations are: ⎟
⎠
⎞
⎜
⎝
⎛
−= ct
c
v
xx' γ , yy' = , zz' =
and
⎟
⎠
⎞
⎜
⎝
⎛
−= x
c
v
ctct' γ
The px, py, pz, and E/c transformation
equations are: ⎟
⎠
⎞
⎜
⎝
⎛
−=
c
E
c
v
p'p xx γ , ,yy p'p = zz p'p =
and
⎟
⎠
⎞
⎜
⎝
⎛
−= xp
c
v
c
E
c
E'
γ
Note that the transformation equations for x, y, z, and ct and the transformation
equations for px, py, pz, and E/c are identical.