This document contains lecture notes from Physics 111 on linear momentum and collisions. It discusses key concepts like conservation of momentum, impulse, elastic and inelastic collisions. For a car crash test example, the impulse delivered to the car is calculated as 64.2 Ns and the average force on the car is 1,764 N directed into the car. Conservation of momentum is used to solve example problems involving collisions between objects on ice.
This document summarizes a physics lecture on linear momentum and collisions. It defines key concepts like linear momentum, impulse, conservation of momentum and different types of collisions. It provides examples of calculating momentum and impulse in one-dimensional collisions. The document also discusses two-dimensional collisions and the conservation of momentum and kinetic energy in elastic and inelastic collisions.
Momentum ;
Impulse;
Conservation of Momentum;
1-D Collisions;
2-D Collisions;
The Center of Mass;
Impact of car;
Impact at intersection;
Conservation of energy;
The 3 conservation laws are:
1) Conservation of energy - the total energy of an isolated system remains constant over time.
2) Conservation of linear momentum - the total momentum of a system remains constant, as long as no external force acts on the system.
3) Conservation of angular momentum - the angular momentum of a system remains constant, as long as no external torque acts on it.
Long 50slideschapter 5 motion notes [autosaved]Duluth Middle
This document summarizes Newton's laws of motion. Newton's first law states that an object at rest stays at rest and an object in motion stays in motion with the same speed and direction unless acted upon by an unbalanced force. Newton's second law relates the net force on an object to its acceleration. Newton's third law states that for every action force there is an equal and opposite reaction force. The document also discusses concepts such as motion, velocity, acceleration, momentum, and conservation of momentum.
This document covers concepts in one-dimensional and three-dimensional kinematics, dynamics, work, energy, momentum, rotational motion, and more. Examples are provided to demonstrate how to apply equations for instantaneous and average velocity/acceleration, projectile motion, Newton's laws, work-energy theorem, impulse-momentum, center of mass, moment of inertia, and torque. Problem-solving strategies are outlined for analyzing forces, energy, momentum, and rotational equilibrium.
The document discusses different types of collisions between objects. In an elastic collision, both momentum and kinetic energy are conserved. In an inelastic collision, momentum is conserved but kinetic energy is not. In a perfectly inelastic collision, momentum is conserved and kinetic energy is not conserved, and the objects stick together after collision and move with the same final velocity. Momentum is conserved in all collision types.
This document summarizes key concepts from Physics 111 lecture 7 on potential energy and energy conservation. It defines work, kinetic energy, gravitational potential energy, elastic potential energy, and conservative and non-conservative forces. It presents the work-energy theorem and its extension to include potential energy. Examples are provided on calculating speed and kinetic energy for a diver and a block projected up an incline using the extended work-energy theorem.
This document summarizes a physics lecture on linear momentum and collisions. It defines key concepts like linear momentum, impulse, conservation of momentum and different types of collisions. It provides examples of calculating momentum and impulse in one-dimensional collisions. The document also discusses two-dimensional collisions and the conservation of momentum and kinetic energy in elastic and inelastic collisions.
Momentum ;
Impulse;
Conservation of Momentum;
1-D Collisions;
2-D Collisions;
The Center of Mass;
Impact of car;
Impact at intersection;
Conservation of energy;
The 3 conservation laws are:
1) Conservation of energy - the total energy of an isolated system remains constant over time.
2) Conservation of linear momentum - the total momentum of a system remains constant, as long as no external force acts on the system.
3) Conservation of angular momentum - the angular momentum of a system remains constant, as long as no external torque acts on it.
Long 50slideschapter 5 motion notes [autosaved]Duluth Middle
This document summarizes Newton's laws of motion. Newton's first law states that an object at rest stays at rest and an object in motion stays in motion with the same speed and direction unless acted upon by an unbalanced force. Newton's second law relates the net force on an object to its acceleration. Newton's third law states that for every action force there is an equal and opposite reaction force. The document also discusses concepts such as motion, velocity, acceleration, momentum, and conservation of momentum.
This document covers concepts in one-dimensional and three-dimensional kinematics, dynamics, work, energy, momentum, rotational motion, and more. Examples are provided to demonstrate how to apply equations for instantaneous and average velocity/acceleration, projectile motion, Newton's laws, work-energy theorem, impulse-momentum, center of mass, moment of inertia, and torque. Problem-solving strategies are outlined for analyzing forces, energy, momentum, and rotational equilibrium.
The document discusses different types of collisions between objects. In an elastic collision, both momentum and kinetic energy are conserved. In an inelastic collision, momentum is conserved but kinetic energy is not. In a perfectly inelastic collision, momentum is conserved and kinetic energy is not conserved, and the objects stick together after collision and move with the same final velocity. Momentum is conserved in all collision types.
This document summarizes key concepts from Physics 111 lecture 7 on potential energy and energy conservation. It defines work, kinetic energy, gravitational potential energy, elastic potential energy, and conservative and non-conservative forces. It presents the work-energy theorem and its extension to include potential energy. Examples are provided on calculating speed and kinetic energy for a diver and a block projected up an incline using the extended work-energy theorem.
This document provides information about different types of collisions, including perfectly elastic, inelastic, and perfectly inelastic collisions. It defines key concepts like momentum and kinetic energy conservation and discusses how to set up and solve problems involving 1D and 2D collisions. Examples are provided to demonstrate solving for unknown velocities after collisions using conservation of momentum and kinetic energy equations. Collisions at intersections and "glancing" collisions where objects move off at angles are also addressed.
This document discusses momentum and impulse in mechanics. It defines momentum as the product of mass and velocity, and impulse as the product of force and time. Impulse causes changes in momentum according to the equation Ft = Δp. Collisions can be elastic, inelastic, or perfectly inelastic, with elastic collisions conserving both momentum and kinetic energy. Examples are provided to demonstrate calculating momentum, velocity, and kinetic energy before and after collisions.
The document provides conceptual problems and their solutions related to Newton's Laws of motion.
1) A problem asks how to determine if a limousine is changing speed or direction using a small object on a string. The solution is that if the string remains vertical, the reference frame is inertial.
2) Another problem asks for two situations where apparent weight in an elevator is greater than true weight. The solution states this occurs when the elevator accelerates upward, either slowing down or speeding up.
3) A third problem involves forces between blocks and identifies which constitute Newton's third law pairs. The normal forces between blocks and between a block and table are identified as third law pairs.
Karen Adelan presented on the topic of classical mechanics and energy. Some key points:
- Energy is a conserved quantity that can change forms but is never created or destroyed. It is useful for describing motion when Newton's laws are difficult to apply.
- Kinetic energy is the energy of motion and depends on an object's mass and speed. The work-kinetic energy theorem states that the net work done on an object equals the change in its kinetic energy.
- Potential energy is the energy an object possesses due to its position or state. The work done by a constant force equals the product of force, displacement, and the cosine of the angle between them.
This document summarizes a physics lecture on potential energy and the conservation of energy. It introduces concepts like work, kinetic energy, gravitational potential energy, elastic potential energy, and the work-energy theorem. It discusses how these concepts can be used to solve problems involving objects moving under conservative and non-conservative forces. Examples are provided, such as calculating the speeds of a diver at different heights, and finding the maximum height of a block projected up an incline.
This document contains a chapter summary for Chapter 7 of Giambattista Physics. It covers the following topics:
1. Linear momentum as a conserved vector quantity. Momentum can be transferred between objects during interactions.
2. The impulse-momentum theorem, which relates impulse (force over time) to changes in momentum. It can be used to analyze collisions.
3. Conservation of momentum, where the total momentum of a system remains constant if the external force is zero.
4. The concept of center of mass as a single point that represents the average location of a system of objects.
The chapter contains examples calculating momentum, impulse, average force, and center of mass for various physical systems
Learn Online Courses of Subject Engineering Mechanics of First Year Engineering. Clear the Concepts of Engineering Mechanics Through Video Lectures and PDF Notes. Visit us: https://ekeeda.com/streamdetails/subject/Engineering-Mechanics
Ekeeda Provides Online Video Lectures for Civil Engineering Degree Subject Courses for All Engineering Universities. Visit us: https://ekeeda.com/streamdetails/stream/civil-engineering
Newton's three laws of motion are introduced. The objectives are to understand how forces affect motion, state the three laws, know when each law applies, and apply Newton's second law of F=ma to problems involving linear motion and systems of connected bodies. Examples are provided to illustrate calculating acceleration, tension, tractive effort, and motion on inclined planes using the three laws of motion.
1. Momentum is defined as the product of an object's mass and velocity. It is a conserved quantity such that the total momentum of an isolated system remains constant.
2. During collisions, conservation of momentum states that the total momentum of colliding objects before the collision equals the total momentum after. If no external forces are applied, momentum is conserved.
3. Collisions can be elastic, where both momentum and kinetic energy are conserved, or inelastic where kinetic energy is not conserved but momentum still is. The analysis of collisions uses conservation laws to solve for unknown velocities.
The document discusses an example involving the linear momentum of three birds with masses mg = 9 kg and md = 4 kg. It provides the equations to calculate the total momentum vector as the sum of the individual momentum vectors and gives the expressions to calculate the magnitude and direction of the total momentum.
- Newton's laws of motion describe the relationship between an object and the forces acting upon it.
- The first law states that an object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force.
- The second law states that the acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object.
This document discusses linear momentum and collisions. It begins by defining key terms like conservation of energy, momentum, impulse, and conservation of momentum. It then discusses one-dimensional and two-dimensional collisions. For 1D collisions, it provides methods for solving problems using conservation of momentum and energy equations. For 2D collisions, it notes that momentum is conserved in all directions and two equations are needed, with kinetic energy conserved for elastic collisions.
This document appears to be an assignment on momentum and collisions submitted by a student named AFGAAB. It includes definitions of key concepts like momentum, impulse, Newton's second law of motion, and the law of conservation of momentum. It also provides examples of calculating momentum and solving problems involving collisions between objects using the conservation of momentum principle. The assignment contains diagrams and solutions to sample problems.
The document discusses key concepts in mechanics including:
1. Free body diagrams show only the external forces acting on an object and are useful for solving dynamics problems.
2. Newton's Second Law states that acceleration is proportional to net force and inversely proportional to mass.
3. Impulse is the product of force and time and equals change in momentum, affecting how objects move after collisions or other impacts.
12. kinetics of particles impulse momentum methodEkeeda
Learn Online Courses of Subject Engineering Mechanics of First Year Engineering. Clear the Concepts of Engineering Mechanics Through Video Lectures and PDF Notes.
https://ekeeda.com/streamdetails/subject/Engineering-Mechanics
This document discusses the impulse-momentum method for solving kinetics problems involving particles. It begins by introducing impulse as the product of force and time. The impulse-momentum equation is derived from Newton's second law, relating total impulse on a particle to the change in its momentum. This equation can be used to find velocities when forces and times are known. The document also discusses the conservation of momentum equation for closed systems where net impulse is zero. Collisions between particles are examined, defining the coefficient of restitution as the ratio of impulse during separation to impulse during contact.
Kinetics of particles impulse momentum methodEkeeda
Ekeeda Provides Online Video Lectures for Civil Engineering Degree Subject Courses for All Engineering Universities. Visit us: https://ekeeda.com/streamdetails/stream/civil-engineering
Newton™s Laws; Moment of a Vector; Gravitation; Finite Rotations; Trajectory of a Projectile with Air Resistance; The Simple Pendulum; The Linear Harmonic Oscillator; The Damped Harmonic Oscillator
JAMES WEBB STUDY THE MASSIVE BLACK HOLE SEEDSSérgio Sacani
The pathway(s) to seeding the massive black holes (MBHs) that exist at the heart of galaxies in the present and distant Universe remains an unsolved problem. Here we categorise, describe and quantitatively discuss the formation pathways of both light and heavy seeds. We emphasise that the most recent computational models suggest that rather than a bimodal-like mass spectrum between light and heavy seeds with light at one end and heavy at the other that instead a continuum exists. Light seeds being more ubiquitous and the heavier seeds becoming less and less abundant due the rarer environmental conditions required for their formation. We therefore examine the different mechanisms that give rise to different seed mass spectrums. We show how and why the mechanisms that produce the heaviest seeds are also among the rarest events in the Universe and are hence extremely unlikely to be the seeds for the vast majority of the MBH population. We quantify, within the limits of the current large uncertainties in the seeding processes, the expected number densities of the seed mass spectrum. We argue that light seeds must be at least 103 to 105 times more numerous than heavy seeds to explain the MBH population as a whole. Based on our current understanding of the seed population this makes heavy seeds (Mseed > 103 M⊙) a significantly more likely pathway given that heavy seeds have an abundance pattern than is close to and likely in excess of 10−4 compared to light seeds. Finally, we examine the current state-of-the-art in numerical calculations and recent observations and plot a path forward for near-future advances in both domains.
Sexuality - Issues, Attitude and Behaviour - Applied Social Psychology - Psyc...PsychoTech Services
A proprietary approach developed by bringing together the best of learning theories from Psychology, design principles from the world of visualization, and pedagogical methods from over a decade of training experience, that enables you to: Learn better, faster!
This document provides information about different types of collisions, including perfectly elastic, inelastic, and perfectly inelastic collisions. It defines key concepts like momentum and kinetic energy conservation and discusses how to set up and solve problems involving 1D and 2D collisions. Examples are provided to demonstrate solving for unknown velocities after collisions using conservation of momentum and kinetic energy equations. Collisions at intersections and "glancing" collisions where objects move off at angles are also addressed.
This document discusses momentum and impulse in mechanics. It defines momentum as the product of mass and velocity, and impulse as the product of force and time. Impulse causes changes in momentum according to the equation Ft = Δp. Collisions can be elastic, inelastic, or perfectly inelastic, with elastic collisions conserving both momentum and kinetic energy. Examples are provided to demonstrate calculating momentum, velocity, and kinetic energy before and after collisions.
The document provides conceptual problems and their solutions related to Newton's Laws of motion.
1) A problem asks how to determine if a limousine is changing speed or direction using a small object on a string. The solution is that if the string remains vertical, the reference frame is inertial.
2) Another problem asks for two situations where apparent weight in an elevator is greater than true weight. The solution states this occurs when the elevator accelerates upward, either slowing down or speeding up.
3) A third problem involves forces between blocks and identifies which constitute Newton's third law pairs. The normal forces between blocks and between a block and table are identified as third law pairs.
Karen Adelan presented on the topic of classical mechanics and energy. Some key points:
- Energy is a conserved quantity that can change forms but is never created or destroyed. It is useful for describing motion when Newton's laws are difficult to apply.
- Kinetic energy is the energy of motion and depends on an object's mass and speed. The work-kinetic energy theorem states that the net work done on an object equals the change in its kinetic energy.
- Potential energy is the energy an object possesses due to its position or state. The work done by a constant force equals the product of force, displacement, and the cosine of the angle between them.
This document summarizes a physics lecture on potential energy and the conservation of energy. It introduces concepts like work, kinetic energy, gravitational potential energy, elastic potential energy, and the work-energy theorem. It discusses how these concepts can be used to solve problems involving objects moving under conservative and non-conservative forces. Examples are provided, such as calculating the speeds of a diver at different heights, and finding the maximum height of a block projected up an incline.
This document contains a chapter summary for Chapter 7 of Giambattista Physics. It covers the following topics:
1. Linear momentum as a conserved vector quantity. Momentum can be transferred between objects during interactions.
2. The impulse-momentum theorem, which relates impulse (force over time) to changes in momentum. It can be used to analyze collisions.
3. Conservation of momentum, where the total momentum of a system remains constant if the external force is zero.
4. The concept of center of mass as a single point that represents the average location of a system of objects.
The chapter contains examples calculating momentum, impulse, average force, and center of mass for various physical systems
Learn Online Courses of Subject Engineering Mechanics of First Year Engineering. Clear the Concepts of Engineering Mechanics Through Video Lectures and PDF Notes. Visit us: https://ekeeda.com/streamdetails/subject/Engineering-Mechanics
Ekeeda Provides Online Video Lectures for Civil Engineering Degree Subject Courses for All Engineering Universities. Visit us: https://ekeeda.com/streamdetails/stream/civil-engineering
Newton's three laws of motion are introduced. The objectives are to understand how forces affect motion, state the three laws, know when each law applies, and apply Newton's second law of F=ma to problems involving linear motion and systems of connected bodies. Examples are provided to illustrate calculating acceleration, tension, tractive effort, and motion on inclined planes using the three laws of motion.
1. Momentum is defined as the product of an object's mass and velocity. It is a conserved quantity such that the total momentum of an isolated system remains constant.
2. During collisions, conservation of momentum states that the total momentum of colliding objects before the collision equals the total momentum after. If no external forces are applied, momentum is conserved.
3. Collisions can be elastic, where both momentum and kinetic energy are conserved, or inelastic where kinetic energy is not conserved but momentum still is. The analysis of collisions uses conservation laws to solve for unknown velocities.
The document discusses an example involving the linear momentum of three birds with masses mg = 9 kg and md = 4 kg. It provides the equations to calculate the total momentum vector as the sum of the individual momentum vectors and gives the expressions to calculate the magnitude and direction of the total momentum.
- Newton's laws of motion describe the relationship between an object and the forces acting upon it.
- The first law states that an object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force.
- The second law states that the acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object.
This document discusses linear momentum and collisions. It begins by defining key terms like conservation of energy, momentum, impulse, and conservation of momentum. It then discusses one-dimensional and two-dimensional collisions. For 1D collisions, it provides methods for solving problems using conservation of momentum and energy equations. For 2D collisions, it notes that momentum is conserved in all directions and two equations are needed, with kinetic energy conserved for elastic collisions.
This document appears to be an assignment on momentum and collisions submitted by a student named AFGAAB. It includes definitions of key concepts like momentum, impulse, Newton's second law of motion, and the law of conservation of momentum. It also provides examples of calculating momentum and solving problems involving collisions between objects using the conservation of momentum principle. The assignment contains diagrams and solutions to sample problems.
The document discusses key concepts in mechanics including:
1. Free body diagrams show only the external forces acting on an object and are useful for solving dynamics problems.
2. Newton's Second Law states that acceleration is proportional to net force and inversely proportional to mass.
3. Impulse is the product of force and time and equals change in momentum, affecting how objects move after collisions or other impacts.
12. kinetics of particles impulse momentum methodEkeeda
Learn Online Courses of Subject Engineering Mechanics of First Year Engineering. Clear the Concepts of Engineering Mechanics Through Video Lectures and PDF Notes.
https://ekeeda.com/streamdetails/subject/Engineering-Mechanics
This document discusses the impulse-momentum method for solving kinetics problems involving particles. It begins by introducing impulse as the product of force and time. The impulse-momentum equation is derived from Newton's second law, relating total impulse on a particle to the change in its momentum. This equation can be used to find velocities when forces and times are known. The document also discusses the conservation of momentum equation for closed systems where net impulse is zero. Collisions between particles are examined, defining the coefficient of restitution as the ratio of impulse during separation to impulse during contact.
Kinetics of particles impulse momentum methodEkeeda
Ekeeda Provides Online Video Lectures for Civil Engineering Degree Subject Courses for All Engineering Universities. Visit us: https://ekeeda.com/streamdetails/stream/civil-engineering
Newton™s Laws; Moment of a Vector; Gravitation; Finite Rotations; Trajectory of a Projectile with Air Resistance; The Simple Pendulum; The Linear Harmonic Oscillator; The Damped Harmonic Oscillator
JAMES WEBB STUDY THE MASSIVE BLACK HOLE SEEDSSérgio Sacani
The pathway(s) to seeding the massive black holes (MBHs) that exist at the heart of galaxies in the present and distant Universe remains an unsolved problem. Here we categorise, describe and quantitatively discuss the formation pathways of both light and heavy seeds. We emphasise that the most recent computational models suggest that rather than a bimodal-like mass spectrum between light and heavy seeds with light at one end and heavy at the other that instead a continuum exists. Light seeds being more ubiquitous and the heavier seeds becoming less and less abundant due the rarer environmental conditions required for their formation. We therefore examine the different mechanisms that give rise to different seed mass spectrums. We show how and why the mechanisms that produce the heaviest seeds are also among the rarest events in the Universe and are hence extremely unlikely to be the seeds for the vast majority of the MBH population. We quantify, within the limits of the current large uncertainties in the seeding processes, the expected number densities of the seed mass spectrum. We argue that light seeds must be at least 103 to 105 times more numerous than heavy seeds to explain the MBH population as a whole. Based on our current understanding of the seed population this makes heavy seeds (Mseed > 103 M⊙) a significantly more likely pathway given that heavy seeds have an abundance pattern than is close to and likely in excess of 10−4 compared to light seeds. Finally, we examine the current state-of-the-art in numerical calculations and recent observations and plot a path forward for near-future advances in both domains.
Sexuality - Issues, Attitude and Behaviour - Applied Social Psychology - Psyc...PsychoTech Services
A proprietary approach developed by bringing together the best of learning theories from Psychology, design principles from the world of visualization, and pedagogical methods from over a decade of training experience, that enables you to: Learn better, faster!
PPT on Alternate Wetting and Drying presented at the three-day 'Training and Validation Workshop on Modules of Climate Smart Agriculture (CSA) Technologies in South Asia' workshop on April 22, 2024.
PPT on Direct Seeded Rice presented at the three-day 'Training and Validation Workshop on Modules of Climate Smart Agriculture (CSA) Technologies in South Asia' workshop on April 22, 2024.
ESA/ACT Science Coffee: Diego Blas - Gravitational wave detection with orbita...Advanced-Concepts-Team
Presentation in the Science Coffee of the Advanced Concepts Team of the European Space Agency on the 07.06.2024.
Speaker: Diego Blas (IFAE/ICREA)
Title: Gravitational wave detection with orbital motion of Moon and artificial
Abstract:
In this talk I will describe some recent ideas to find gravitational waves from supermassive black holes or of primordial origin by studying their secular effect on the orbital motion of the Moon or satellites that are laser ranged.
Signatures of wave erosion in Titan’s coastsSérgio Sacani
The shorelines of Titan’s hydrocarbon seas trace flooded erosional landforms such as river valleys; however, it isunclear whether coastal erosion has subsequently altered these shorelines. Spacecraft observations and theo-retical models suggest that wind may cause waves to form on Titan’s seas, potentially driving coastal erosion,but the observational evidence of waves is indirect, and the processes affecting shoreline evolution on Titanremain unknown. No widely accepted framework exists for using shoreline morphology to quantitatively dis-cern coastal erosion mechanisms, even on Earth, where the dominant mechanisms are known. We combinelandscape evolution models with measurements of shoreline shape on Earth to characterize how differentcoastal erosion mechanisms affect shoreline morphology. Applying this framework to Titan, we find that theshorelines of Titan’s seas are most consistent with flooded landscapes that subsequently have been eroded bywaves, rather than a uniform erosional process or no coastal erosion, particularly if wave growth saturates atfetch lengths of tens of kilometers.
Describing and Interpreting an Immersive Learning Case with the Immersion Cub...Leonel Morgado
Current descriptions of immersive learning cases are often difficult or impossible to compare. This is due to a myriad of different options on what details to include, which aspects are relevant, and on the descriptive approaches employed. Also, these aspects often combine very specific details with more general guidelines or indicate intents and rationales without clarifying their implementation. In this paper we provide a method to describe immersive learning cases that is structured to enable comparisons, yet flexible enough to allow researchers and practitioners to decide which aspects to include. This method leverages a taxonomy that classifies educational aspects at three levels (uses, practices, and strategies) and then utilizes two frameworks, the Immersive Learning Brain and the Immersion Cube, to enable a structured description and interpretation of immersive learning cases. The method is then demonstrated on a published immersive learning case on training for wind turbine maintenance using virtual reality. Applying the method results in a structured artifact, the Immersive Learning Case Sheet, that tags the case with its proximal uses, practices, and strategies, and refines the free text case description to ensure that matching details are included. This contribution is thus a case description method in support of future comparative research of immersive learning cases. We then discuss how the resulting description and interpretation can be leveraged to change immersion learning cases, by enriching them (considering low-effort changes or additions) or innovating (exploring more challenging avenues of transformation). The method holds significant promise to support better-grounded research in immersive learning.
(June 12, 2024) Webinar: Development of PET theranostics targeting the molecu...Scintica Instrumentation
Targeting Hsp90 and its pathogen Orthologs with Tethered Inhibitors as a Diagnostic and Therapeutic Strategy for cancer and infectious diseases with Dr. Timothy Haystead.
The cost of acquiring information by natural selectionCarl Bergstrom
This is a short talk that I gave at the Banff International Research Station workshop on Modeling and Theory in Population Biology. The idea is to try to understand how the burden of natural selection relates to the amount of information that selection puts into the genome.
It's based on the first part of this research paper:
The cost of information acquisition by natural selection
Ryan Seamus McGee, Olivia Kosterlitz, Artem Kaznatcheev, Benjamin Kerr, Carl T. Bergstrom
bioRxiv 2022.07.02.498577; doi: https://doi.org/10.1101/2022.07.02.498577
2. May 23, 2023
Linear Momentum
and Collisions
Conservation
of Energy
Momentum
Impulse
Conservation
of Momentum
1-D Collisions
2-D Collisions
The Center of Mass
3. May 23, 2023
Conservation of Energy
D E = D K + D U = 0 if conservative forces are the only
forces that do work on the system.
The total amount of energy in the system is constant.
D E = D K + D U = -fkd if friction forces are doing work
on the system.
The total amount of energy in the system is still
constant, but the change in mechanical energy goes
into “internal energy” or heat.
2
2
2
2
2
1
2
1
2
1
2
1
i
i
i
f
f
f kx
mgy
mv
kx
mgy
mv
2
2
2
2
2
1
2
1
2
1
2
1
i
i
i
f
f
f
k kx
mgy
mv
kx
mgy
mv
d
f
4. May 23, 2023
Linear Momentum
This is a new fundamental quantity, like force, energy.
It is a vector quantity (points in same direction as
velocity).
The linear momentum p of an object of mass m moving
with a velocity v is defined to be the product of the
mass and velocity:
The terms momentum and linear momentum will be
used interchangeably in the text
Momentum depend on an object’s mass and velocity
v
m
p
5. May 23, 2023
Momentum and Energy
Two objects with masses m1 and m2
have equal kinetic energy. How do the
magnitudes of their momenta
compare?
(A) p1 < p2
(B) p1 = p2
(C) p1 > p2
(D) Not enough information is given
6. May 23, 2023
Linear Momentum, cont’d
Linear momentum is a vector quantity
Its direction is the same as the direction of
the velocity
The dimensions of momentum are ML/T
The SI units of momentum are kg m / s
Momentum can be expressed in component
form:
px = mvx py = mvy pz = mvz
m
p v
7. May 23, 2023
Newton’s Law and Momentum
Newton’s Second Law can be used to relate the
momentum of an object to the resultant force
acting on it
The change in an object’s momentum divided by
the elapsed time equals the constant net force
acting on the object
t
v
m
t
v
m
a
m
Fnet
D
D
D
D
)
(
net
F
t
p
D
D
interval
time
momentum
in
change
8. May 23, 2023
Impulse
When a single, constant force acts on the
object, there is an impulse delivered to the
object
is defined as the impulse
The equality is true even if the force is not constant
Vector quantity, the direction is the same as the
direction of the force
I
t
F
I D
net
F
t
p
D
D
interval
time
momentum
in
change
9. May 23, 2023
Impulse-Momentum Theorem
The theorem states
that the impulse
acting on a system is
equal to the change
in momentum of the
system
i
f v
m
v
m
p
I
D
I
t
F
p net
D
D
10. May 23, 2023
Calculating the Change of Momentum
0 ( )
p m v mv
D
( )
after before
after before
after before
p p p
mv mv
m v v
D
For the teddy bear
For the bouncing ball
( ) 2
p m v v mv
D
11. May 23, 2023
How Good Are the Bumpers?
In a crash test, a car of mass 1.5103 kg collides with a wall and
rebounds as in figure. The initial and final velocities of the car are vi=-15
m/s and vf = 2.6 m/s, respectively. If the collision lasts for 0.15 s, find
(a) the impulse delivered to the car due to the collision
(b) the size and direction of the average force exerted on the car
12. May 23, 2023
How Good Are the Bumpers?
In a crash test, a car of mass 1.5103 kg collides with a wall and
rebounds as in figure. The initial and final velocities of the car are vi=-15
m/s and vf = 2.6 m/s, respectively. If the collision lasts for 0.15 s, find
(a) the impulse delivered to the car due to the collision
(b) the size and direction of the average force exerted on the car
s
m
kg
s
m
kg
mv
p i
i /
10
25
.
2
)
/
15
)(
10
5
.
1
( 4
3
N
s
s
m
kg
t
I
t
p
Fav
5
4
10
76
.
1
15
.
0
/
10
64
.
2
D
D
D
s
m
kg
s
m
kg
mv
p f
f /
10
39
.
0
)
/
6
.
2
)(
10
5
.
1
( 4
3
s
m
kg
s
m
kg
s
m
kg
mv
mv
p
p
I i
f
i
f
/
10
64
.
2
)
/
10
25
.
2
(
)
/
10
39
.
0
(
4
4
4
13. May 23, 2023
Impulse-Momentum Theorem
A child bounces a 100 g superball on the
sidewalk. The velocity of the superball
changes from 10 m/s downward to 10 m/s
upward. If the contact time with the
sidewalk is 0.1s, what is the magnitude of
the impulse imparted to the superball?
(A) 0
(B) 2 kg-m/s
(C) 20 kg-m/s
(D) 200 kg-m/s
(E) 2000 kg-m/s
i
f v
m
v
m
p
I
D
14. May 23, 2023
Impulse-Momentum Theorem 2
A child bounces a 100 g superball on the
sidewalk. The velocity of the superball
changes from 10 m/s downward to 10 m/s
upward. If the contact time with the
sidewalk is 0.1s, what is the magnitude of
the force between the sidewalk and the
superball?
(A) 0
(B) 2 N
(C) 20 N
(D) 200 N
(E) 2000 N
t
v
m
v
m
t
p
t
I
F
i
f
D
D
D
D
15. May 23, 2023
Conservation of Momentum
In an isolated and closed system,
the total momentum of the
system remains constant in time.
Isolated system: no external forces
Closed system: no mass enters or
leaves
The linear momentum of each
colliding body may change
The total momentum P of the
system cannot change.
16. May 23, 2023
Conservation of Momentum
Start from impulse-momentum
theorem
Since
Then
So
i
f v
m
v
m
t
F 2
2
2
2
12
D
i
f v
m
v
m
t
F 1
1
1
1
21
D
t
F
t
F D
D 12
21
)
( 2
2
2
2
1
1
1
1 i
f
i
f v
m
v
m
v
m
v
m
f
f
i
i v
m
v
m
v
m
v
m 2
2
1
1
2
2
1
1
17. May 23, 2023
Conservation of Momentum
When no external forces act on a system consisting of
two objects that collide with each other, the total
momentum of the system remains constant in time
When then
For an isolated system
Specifically, the total momentum before the collision will
equal the total momentum after the collision
i
f
net p
p
p
t
F
D
D
0
Dp
0
net
F
i
f p
p
f
f
i
i v
m
v
m
v
m
v
m 2
2
1
1
2
2
1
1
18. May 23, 2023
The Archer
An archer stands at rest on frictionless ice and fires a 0.5-kg arrow
horizontally at 50.0 m/s. The combined mass of the archer and bow is
60.0 kg. With what velocity does the archer move across the ice after
firing the arrow?
f
f
i
i v
m
v
m
v
m
v
m 2
2
1
1
2
2
1
1
f
i p
p
?
,
/
50
,
0
,
5
.
0
,
0
.
60 1
2
2
1
2
1
f
f
i
i v
s
m
v
v
v
kg
m
kg
m
f
f v
m
v
m 2
2
1
1
0
s
m
s
m
kg
kg
v
m
m
v f
f /
417
.
0
)
/
0
.
50
(
0
.
60
5
.
0
2
1
2
1
19. May 23, 2023
Conservation of Momentum
A 100 kg man and 50 kg woman on ice
skates stand facing each other. If the woman
pushes the man backwards so that his final
speed is 1 m/s, at what speed does she recoil?
(A) 0
(B) 0.5 m/s
(C) 1 m/s
(D) 1.414 m/s
(E) 2 m/s
20. May 23, 2023
Types of Collisions
Momentum is conserved in any collision
Inelastic collisions: rubber ball and hard ball
Kinetic energy is not conserved
Perfectly inelastic collisions occur when the objects
stick together
Elastic collisions: billiard ball
both momentum and kinetic energy are conserved
Actual collisions
Most collisions fall between elastic and perfectly
inelastic collisions
21. May 23, 2023
Collisions Summary
In an elastic collision, both momentum and kinetic
energy are conserved
In a non-perfect inelastic collision, momentum is
conserved but kinetic energy is not. Moreover, the
objects do not stick together
In a perfectly inelastic collision, momentum is conserved,
kinetic energy is not, and the two objects stick together
after the collision, so their final velocities are the same
Elastic and perfectly inelastic collisions are limiting cases,
most actual collisions fall in between these two types
Momentum is conserved in all collisions
22. May 23, 2023
More about Perfectly Inelastic
Collisions
When two objects stick together
after the collision, they have
undergone a perfectly inelastic
collision
Conservation of momentum
Kinetic energy is NOT conserved
f
i
i v
m
m
v
m
v
m )
( 2
1
2
2
1
1
2
1
2
2
1
1
m
m
v
m
v
m
v i
i
f
23. May 23, 2023
An SUV Versus a Compact
An SUV with mass 1.80103 kg is travelling eastbound at
+15.0 m/s, while a compact car with mass 9.00102 kg
is travelling westbound at -15.0 m/s. The cars collide
head-on, becoming entangled.
(a) Find the speed of the entangled
cars after the collision.
(b) Find the change in the velocity
of each car.
(c) Find the change in the kinetic
energy of the system consisting
of both cars.
24. May 23, 2023
(a) Find the speed of the entangled
cars after the collision.
f
i
i v
m
m
v
m
v
m )
( 2
1
2
2
1
1
f
i p
p
s
m
v
kg
m
s
m
v
kg
m
i
i
/
15
,
10
00
.
9
/
15
,
10
80
.
1
2
2
2
1
3
1
2
1
2
2
1
1
m
m
v
m
v
m
v i
i
f
s
m
vf /
00
.
5
An SUV Versus a Compact
25. May 23, 2023
(b) Find the change in the velocity
of each car.
s
m
v
v
v i
f /
0
.
10
1
1
D
s
m
v
kg
m
s
m
v
kg
m
i
i
/
15
,
10
00
.
9
/
15
,
10
80
.
1
2
2
2
1
3
1
s
m
vf /
00
.
5
An SUV Versus a Compact
s
m
v
v
v i
f /
0
.
20
2
2
D
s
m
kg
v
v
m
v
m i
f /
10
8
.
1
)
( 4
1
1
1
1
D
0
2
2
1
1
D
D v
m
v
m
s
m
kg
v
v
m
v
m i
f /
10
8
.
1
)
( 4
2
2
2
2
D
26. May 23, 2023
(c) Find the change in the kinetic
energy of the system consisting
of both cars.
J
v
m
v
m
KE i
i
i
5
2
2
2
2
1
1 10
04
.
3
2
1
2
1
s
m
v
kg
m
s
m
v
kg
m
i
i
/
15
,
10
00
.
9
/
15
,
10
80
.
1
2
2
2
1
3
1
s
m
vf /
00
.
5
An SUV Versus a Compact
J
KE
KE
KE i
f
5
10
70
.
2
D
J
v
m
v
m
KE f
f
f
4
2
2
2
2
1
1 10
38
.
3
2
1
2
1
27. May 23, 2023
More About Elastic Collisions
Both momentum and kinetic energy
are conserved
Typically have two unknowns
Momentum is a vector quantity
Direction is important
Be sure to have the correct signs
Solve the equations simultaneously
2
2
2
2
1
1
2
2
2
2
1
1
2
2
1
1
2
2
1
1
2
1
2
1
2
1
2
1
f
f
i
i
f
f
i
i
v
m
v
m
v
m
v
m
v
m
v
m
v
m
v
m
28. May 23, 2023
Elastic Collisions
A simpler equation can be used in place of the KE
equation
i
f
f
i v
v
v
v 2
2
1
1
)
v
v
(
v
v f
2
f
1
i
2
i
1
2
2
2
2
1
1
2
2
2
2
1
1
2
1
2
1
2
1
2
1
f
f
i
i v
m
v
m
v
m
v
m
)
)(
(
)
)(
( 2
2
2
2
2
1
1
1
1
1 i
f
i
f
f
i
f
i v
v
v
v
m
v
v
v
v
m
)
(
)
( 2
2
2
1
1
1 i
f
f
i v
v
m
v
v
m
)
(
)
( 2
2
2
2
2
2
1
2
1
1 i
f
f
i v
v
m
v
v
m
f
f
i
i v
m
v
m
v
m
v
m 2
2
1
1
2
2
1
1
f
f
i
i v
m
v
m
v
m
v
m 2
2
1
1
2
2
1
1
29. May 23, 2023
Summary of Types of Collisions
In an elastic collision, both momentum and kinetic
energy are conserved
In an inelastic collision, momentum is conserved but
kinetic energy is not
In a perfectly inelastic collision, momentum is conserved,
kinetic energy is not, and the two objects stick together
after the collision, so their final velocities are the same
i
f
f
i v
v
v
v 2
2
1
1
f
f
i
i v
m
v
m
v
m
v
m 2
2
1
1
2
2
1
1
f
f
i
i v
m
v
m
v
m
v
m 2
2
1
1
2
2
1
1
f
i
i v
m
m
v
m
v
m )
( 2
1
2
2
1
1
30. May 23, 2023
Conservation of Momentum
An object of mass m moves to the right with a
speed v. It collides head-on with an object of
mass 3m moving with speed v/3 in the opposite
direction. If the two objects stick together, what
is the speed of the combined object, of mass 4m,
after the collision?
(A) 0
(B) v/2
(C) v
(D) 2v
(E) 4v
31. May 23, 2023
Problem Solving for 1D Collisions, 1
Coordinates: Set up a
coordinate axis and define
the velocities with respect
to this axis
It is convenient to make
your axis coincide with one
of the initial velocities
Diagram: In your sketch,
draw all the velocity
vectors and label the
velocities and the masses
32. May 23, 2023
Problem Solving for 1D Collisions, 2
Conservation of
Momentum: Write a
general expression for the
total momentum of the
system before and after
the collision
Equate the two total
momentum expressions
Fill in the known values
f
f
i
i v
m
v
m
v
m
v
m 2
2
1
1
2
2
1
1
33. May 23, 2023
Problem Solving for 1D Collisions, 3
Conservation of Energy:
If the collision is elastic,
write a second equation
for conservation of KE, or
the alternative equation
This only applies to perfectly
elastic collisions
Solve: the resulting
equations simultaneously
i
f
f
i v
v
v
v 2
2
1
1
35. May 23, 2023
Two-Dimensional Collisions
For a general collision of two objects in two-
dimensional space, the conservation of momentum
principle implies that the total momentum of the
system in each direction is conserved
fy
fy
iy
iy
fx
fx
ix
ix
v
m
v
m
v
m
v
m
v
m
v
m
v
m
v
m
2
2
1
1
2
2
1
1
2
2
1
1
2
2
1
1
36. May 23, 2023
Two-Dimensional Collisions
The momentum is conserved in all directions
Use subscripts for
Identifying the object
Indicating initial or final values
The velocity components
If the collision is elastic, use conservation of
kinetic energy as a second equation
Remember, the simpler equation can only be used
for one-dimensional situations
fy
fy
iy
iy
fx
fx
ix
ix
v
m
v
m
v
m
v
m
v
m
v
m
v
m
v
m
2
2
1
1
2
2
1
1
2
2
1
1
2
2
1
1
i
f
f
i v
v
v
v 2
2
1
1
37. May 23, 2023
Glancing Collisions
The “after” velocities have x and y components
Momentum is conserved in the x direction and in the
y direction
Apply conservation of momentum separately to each
direction
fy
fy
iy
iy
fx
fx
ix
ix
v
m
v
m
v
m
v
m
v
m
v
m
v
m
v
m
2
2
1
1
2
2
1
1
2
2
1
1
2
2
1
1
38. May 23, 2023
2-D Collision, example
Particle 1 is moving at
velocity and
particle 2 is at rest
In the x-direction, the
initial momentum is
m1v1i
In the y-direction, the
initial momentum is 0
1i
v
39. May 23, 2023
2-D Collision, example cont
After the collision, the
momentum in the x-direction is
m1v1f cos q m2v2f cos f
After the collision, the
momentum in the y-direction is
m1v1f sin q m2v2f sin f
If the collision is elastic, apply
the kinetic energy equation
f
q
f
q
sin
sin
0
0
cos
cos
0
2
2
1
1
2
2
1
1
1
1
f
f
f
f
i
v
m
v
m
v
m
v
m
v
m
2
2
2
2
1
1
2
1
1
2
1
2
1
2
1
f
f
i v
m
v
m
v
m
40. May 23, 2023
Collision at an Intersection
A car with mass 1.5×103 kg traveling
east at a speed of 25 m/s collides at
an intersection with a 2.5×103 kg van
traveling north at a speed of 20 m/s.
Find the magnitude and direction of
the velocity of the wreckage after the
collision, assuming that the vehicles
undergo a perfectly inelastic collision
and assuming that friction between the
vehicles and the road can be
neglected.
?
?
,
/
20
,
/
25
10
5
.
2
,
10
5
.
1 3
3
q
f
viy
cix
v
c
v
s
m
v
s
m
v
kg
m
kg
m
41. May 23, 2023
Collision at an Intersection
?
?
,
m/s
20
,
m/s
25
kg
10
5
.
2
,
kg
10
5
.
1 3
3
q
f
viy
cix
v
c
v
v
v
m
m
m/s
kg
10
75
.
3 4
cix
c
vix
v
cix
c
xi v
m
v
m
v
m
p
q
cos
)
( f
v
c
vfx
v
cfx
c
xf v
m
m
v
m
v
m
p
q
cos
)
kg
10
00
.
4
(
m/s
kg
10
75
.
3 3
4
f
v
m/s
kg
10
00
.
5 4
viy
v
viy
v
ciy
c
yi v
m
v
m
v
m
p
q
sin
)
( f
v
c
vfy
v
cfy
c
yf v
m
m
v
m
v
m
p
q
sin
)
kg
10
00
.
4
(
m/s
kg
10
00
.
5 3
4
f
v
42. May 23, 2023
Collision at an Intersection
?
?
,
/
20
,
/
25
10
5
.
2
,
10
5
.
1 3
3
q
f
viy
cix
v
c
v
s
m
v
s
m
v
kg
m
kg
m
33
.
1
/
10
75
.
3
/
10
00
.
5
tan 4
4
s
m
kg
s
m
kg
q
1
.
53
)
33
.
1
(
tan 1
q
m/s
6
.
15
1
.
53
sin
)
kg
10
00
.
4
(
m/s
kg
10
00
.
5
3
4
f
v
q
cos
)
kg
10
00
.
4
(
m/s
kg
10
75
.
3 3
4
f
v
q
sin
)
kg
10
00
.
4
(
m/s
kg
10
00
.
5 3
4
f
v
43. May 23, 2023
The Center of Mass
How should we define
the position of the
moving body ?
What is y for Ug =
mgy ?
Take the average
position of mass. Call
“Center of Mass”
(COM or CM)
44. May 23, 2023
The Center of Mass
There is a special point in a system or
object, called the center of mass, that
moves as if all of the mass of the system
is concentrated at that point
The CM of an object or a system is the
point, where the object or the system can
be balanced in the uniform gravitational
field
45. May 23, 2023
The Center of Mass
The center of mass of any symmetric object lies on an
axis of symmetry and on any plane of symmetry
If the object has uniform density
The CM may reside inside the body, or outside the body
46. May 23, 2023
Where is the Center of Mass ?
The center of mass of particles
Two bodies in 1 dimension
2
1
2
2
1
1
m
m
x
m
x
m
xCM
48. May 23, 2023
Where is the Center of Mass ?
Assume m1 = 1 kg, m2 = 3 kg, and x1 =
1 m, x2 = 5 m, where is the center of
mass of these two objects?
A) xCM = 1 m
B) xCM = 2 m
C) xCM = 3 m
D) xCM = 4 m
E) xCM = 5 m
2
1
2
2
1
1
m
m
x
m
x
m
xCM
49. May 23, 2023
Center of Mass
for a System of Particles
Two bodies and one dimension
General case: n bodies and three dimension
where M = m1 + m2 + m3 +…
50. May 23, 2023
Sample Problem : Three particles of masses m1 = 1.2 kg,
m2 = 2.5 kg, and m3 = 3.4 kg form an equilateral triangle of
edge length a = 140 cm. Where is the center of mass of this
system? (Hint: m1 is at (0,0), m2 is at (140 cm,0), and m3 is
at (70 cm, 120 cm), as shown in the figure below.)
3
2
1
3
3
2
2
1
1
1
1
m
m
m
x
m
x
m
x
m
x
m
M
x
n
i
i
i
CM
3
2
1
3
3
2
2
1
1
1
1
m
m
m
y
m
y
m
y
m
y
m
M
y
n
i
i
i
CM
cm
5
.
57
and
cm
82.8
CM
CM y
x
51. May 23, 2023
Motion of a System of Particles
Assume the total mass, M, of the system
remains constant
We can describe the motion of the system
in terms of the velocity and acceleration of
the center of mass of the system
We can also describe the momentum of
the system and Newton’s Second Law for
the system
52. May 23, 2023
Velocity and Momentum of a
System of Particles
The velocity of the center of mass of a system of
particles is
The momentum can be expressed as
The total linear momentum of the system equals
the total mass multiplied by the velocity of the
center of mass
CM
CM
1
i i
i
d
m
dt M
r
v v
CM tot
i i i
i i
M m
v v p p
53. May 23, 2023
Acceleration and Force of the
Center of Mass
The acceleration of the center of mass can be
found by differentiating the velocity with respect
to time
The acceleration can be related to a force
If we sum over all the internal forces, they
cancel in pairs and the net force on the system
is caused only by the external forces
CM
CM
1
i i
i
d
m
dt M
v
a a
CM i
i
M
a F
54. May 23, 2023
Newton’s Second Law
for a System of Particles
Since the only forces are external, the net
external force equals the total mass of the
system multiplied by the acceleration of the
center of mass:
The center of mass of a system of particles of
combined mass M moves like an equivalent
particle of mass M would move under the
influence of the net external force on the system
ext CM
M
F a