Rich Mathematical Problems in Astronomy
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This document contains a powerpoint presentation on using Newton's laws of motion and universal gravitation to derive formulas for orbital velocity, escape velocity, and calculating planetary masses. It includes examples of using these formulas to calculate the mass of Earth from the moon's orbit and estimating escape velocity of Earth. The presentation is intended for pre-AP Algebra 1 or Algebra 2 students to practice manipulating equations and unit conversions. It recommends the Space Math website as a source for additional astronomy math problems.
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Einstein's theories of relativity include the special and general theories. The special theory, published in 1905, deals with inertial frames of reference and the constancy of the speed of light. The general theory, published in 1915, extends these concepts to accelerated frames and explains gravity as a consequence of spacetime curvature. Key findings include spacetime being dynamic, light bending in gravity, and the equivalence of mass and energy. The Michelson-Morley experiment's null results disproved the ether hypothesis and supported Einstein's postulate that the speed of light is independent of motion.
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To move into a higher velocity orbit, you need to fire the satellite's thrusters forwards. Firing forwards does negative work, which decreases the potential energy and increases the kinetic energy, resulting in a higher orbital velocity.
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3) For superconductors, the energy gap Δ must be exceeded before tunneling can occur. Tunneling between two superconductors only allows current when a bias exceeds Δ/e, while tunneling between a normal metal and superconductor begins when the bias reaches ±Δ/e.
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The document discusses Albert Einstein's theories of special and general relativity. Special relativity introduced concepts like the constancy of the speed of light and reference frames. General relativity holds that gravity is not a force but a consequence of objects warping the geometry of spacetime. It predicts phenomena like light deflection near massive objects and gravitational time dilation, which were later confirmed. The document also provides background on the Michelson-Morley experiment and quotes from Einstein and Hawking about the implications of relativity theory.
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1) It showed that Newton's ideas of absolute space and time were incorrect and implied that matter and energy are interconvertible.
2) It established two postulates - the laws of physics apply in all inertial frames, and the speed of light is constant in all frames.
3) This leads to effects like time dilation and length contraction, as measurements of space and time differ for observers in different inertial frames moving relative to one another.
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The document provides information about the history and exploration of space. It discusses some of the first achievements in space such as the first dog (Laika) and humans (Yuri Gagarin and Valentina Tereshkova) in space. It also mentions some of the early rockets launched by India and achievements of the first Indian woman in space, Kalpana Chawla. The document concludes with some facts about space and pros and cons of space exploration.
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Paper Question No. (1)
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Gravitation laws.ppt
- Newton's law of universal gravitation states that every particle in the universe attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
- G is the gravitational constant. Gravity causes objects with mass to accelerate towards one another, as seen with the apple falling towards Earth.
- Kepler's laws describe the motion of planets orbiting the Sun, including that their orbits are ellipses with the Sun at one focus.
Write up Final
1) The document describes a mathematical model for predicting the reentry and impact of satellites into Earth's atmosphere. It uses Newton's second law of motion and models the forces of gravity and drag.
2) By solving the resulting differential equation numerically, the model can determine the impact time and velocity of a satellite. This allows estimates of the damage caused and time for evacuation if the satellite were to impact a populated area.
3) The document provides an example application of the model to Sputnik 1 and graphs the results, showing a impact time of about 5.5 minutes and velocity of 47 m/s.
Important points on Gravitation for JEE Main 2015
1. Newton's law of universal gravitation describes the gravitational attraction between two bodies, with the force of attraction directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
2. The acceleration due to gravity at the earth's surface (g) is calculated using the mass of the earth and its radius. The value of g decreases with increasing height or depth from the earth's surface.
3. Kepler's laws describe the motion of planets orbiting the sun, including that planets move in elliptical orbits with the sun at one focus, a line connecting a planet and the sun sweeps out equal areas in equal time intervals, and the square of the orbital periods are
Extra Practice Questions for Gravitation
1. The document contains multiple choice and structured questions about gravitation.
2. Questions cover topics like gravitational field strength at different heights, gravitational forces between celestial bodies, and equations for orbital motion.
3. Sample problems are worked out relating to gravitational forces, escape velocities, and the distribution of molecular speeds as it relates to atmospheric evolution.
PHYSICS CLASS XI Chapter 5 - gravitation
Here are the key steps to calculate the mass of Earth (M) from the given data:
1) Acceleration due to gravity on Earth's surface (g) = 9.81 m/s^2
2) Universal gravitational constant (G) = 6.67x10^-11 Nm^2/kg^2
3) Radius of Earth (R) = 6.37x10^6 m
4) Using the formula for acceleration due to gravity:
g = GM/R^2
5) Rearranging the terms:
M = gR^2/G
6) Substituting the values:
M = (9.81 m/s^2)(
chapter5-gravitationppt-copy-211229151431 (2).pdf
Here are the key steps to calculate the mass of Earth (M) from the given data:
1) Acceleration due to gravity on Earth's surface (g) = 9.81 m/s^2
2) Universal gravitational constant (G) = 6.67x10^-11 Nm^2/kg^2
3) Radius of Earth (R) = 6.37x10^6 m
4) Using the formula for acceleration due to gravity:
g = GM/R^2
5) Rearranging the terms:
M = gR^2/G
6) Substituting the values:
M = (9.81 m/s^2)(
Why Does the Atmosphere Rotate? Trajectory of a desorbed molecule
As a step toward understanding why the Earth's atmosphere "rotates" with the Earth, we use using Geometric (Clifford) Algebra to investigate the trajectory of a single molecule that desorbs vertically upward from the Equator, then falls back to Earth without colliding with any other molecules. Sample calculations are presented for a molecule whose vertical velocity is equal to the surface velocity of the Earth at the Equator (463 m/s) and for one with a vertical velocity three times as high. The latter velocity is sufficient for the molecule to reach the Kármán Line (100,000 m). We find that both molecules fall to Earth behind the point from which they desorbed: by 0.25 degrees of latitude for the higher vertical velocity, but by only 0.001 degrees for the lower.
Gravitation - Physics
This document discusses concepts related to gravitation and gravity. It begins with a brief history of gravity and Newton's law of gravitation. It then defines gravitation as the attractive force between any two objects with mass, and defines gravity as the gravitational force that occurs between Earth and other bodies. The key points are that gravitation is a universal force, while gravity specifically refers to the gravitational attraction of Earth. The document goes on to provide explanations and formulas for concepts like gravitational constant, acceleration due to gravity, mass vs weight, and center of gravity vs center of mass.
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* Use Newton's Law of Cooling
* Use logistic-growth models
* Choose an appropriate model for data
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* Identify the circumcenter and incenter of a triangle
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* Identify, write, and analyze conditional statements
* Write the inverse, converse, and contrapositive of a conditional statement
* Write a counterexample to a fake conjecture
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* Find the midpoint of two given points
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1.5 Quadratic Equations.pdf
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* Connect functions to their graphs
* Graph piecewise-defined functions
* Graph absolute value functions
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* Interpret graphs
* Use the vertical line test to determine a function
3.1 Functions
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* Develop skills in constructing and interpreting the graphs of functions
* Learn to apply this knowledge in a variety of situations
2.5 Transformations of Functions
* Recognize graphs of common functions.
* Graph functions using vertical and horizontal shifts.
* Graph functions using reflections about the x-axis and the y-axis.
* Graph functions using compressions and stretches.
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2.2 More on Functions and Their Graphs
This document discusses functions and their graphs. It defines increasing, decreasing and constant functions based on how the function values change as the input increases. Relative maxima and minima are points where a function changes from increasing to decreasing. Symmetry of functions is classified by the y-axis, x-axis and origin. Even functions are symmetric about the y-axis, odd functions are symmetric about the origin. Piecewise functions have different definitions over different intervals.
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This document provides examples and steps for solving various types of equations beyond linear equations, including:
1) Polynomial equations solved by factoring
2) Equations with radicals where radicals are eliminated by raising both sides to a power
3) Equations with rational exponents where both sides are raised to the reciprocal power
4) Equations quadratic in form where an algebraic substitution is made to transform into a quadratic equation
5) Absolute value equations where both positive and negative solutions must be considered.
1.5 Quadratic Equations (Review)
This document provides instruction on factoring quadratic equations. It begins by reviewing factoring polynomials and trinomials. It then discusses factoring binomials using difference of squares, sum/difference of cubes, and other patterns. Finally, it explains that a quadratic equation can be solved by factoring if it can be written as a product of two linear factors. An example demonstrates factoring a quadratic equation by finding the two values that make each factor equal to zero.
2.1 Basics of Functions and Their Graphs
This document provides an overview of functions and their graphs. It defines what constitutes a function, discusses domain and range, and how to identify functions using the vertical line test. Key points covered include:
- A function is a relation where each input has a single, unique output
- The domain is the set of inputs and the range is the set of outputs
- Functions can be represented by ordered pairs, graphs, or equations
- The vertical line test identifies functions as those where a vertical line intersects the graph at most once
- Intercepts occur where the graph crosses the x or y-axis
9.6 Binomial Theorem
The document discusses the binomial theorem, which provides a formula for expanding binomial expressions of the form (a + b)^n. It gives the formula for finding the coefficient of the term containing b^r as nCr. Several examples are worked out applying the binomial theorem to expand binomial expressions and find specific terms. Factorial notation is introduced for writing the coefficients. The document also discusses using calculators and Desmos to evaluate binomial coefficients. Practice problems are assigned from previous sections.
13.3 Venn Diagrams & Two-Way Tables
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13.2 Independent & Dependent Events
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* Solve counting problems using permutations involving n distinct objects.
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9.4 Series and Their Notations
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Rich Mathematical Problems in Astronomy
- 1. Sandra Miller and Stephanie Smith Lamar High School Arlington, TX
- 3. This problem is designed to occur during a Geometry unit on circles. A line tangent to a circle forms a right angle with a radius drawn at the point of tangency.
- 4. d r h r r – radius of the planet/moon h – height of the observer (eyes) d – distance to the horizon
- 5. d r h r r – radius of the planet/moon h – height of the observer (eyes) d – distance to the horizon d= r + h) − r 2 ( 2 d = r 2 + 2rh + h2 − r 2 d = h ( 2r + h )
- 6. Object Radius Horizon Earth 3959 mi. 3 mi. Moon 1080 mi. Mars 2106 mi. Jupiter 43,441 mi.
- 7. Object Radius Horizon Earth 3959 mi. 3 mi. Moon 1080 mi. 1.6 mi. Mars 2106 mi. 2.2 mi. Jupiter 43,441 mi. 9.9 mi.
- 9. This problem set is geared toward a Pre-AP Algebra I class or an Algebra II class. By working through this packet, a student will practice Simplifying literal equations Creating formulas Unit conversions Using formulas to solve problems
- 10. Sir Isaac Newton developed three equations that we will use to develop some interesting information about the solar system. F = ma When a force F acts on a body of mass m, it produces in it an acceleration a equal to the force divided by the mass. v a= r The centripetal acceleration a of any body moving in a circular orbit is equal to the square of its velocity v divided by the radius r of the orbit. Gm1m2 F= r2 The grativational force F between two objects is proportional to the product of their two masses, divided by the distance between them. 2
- 11. If we substitute the formula for centripetal acceleration into the F = ma equation, we have an equation for the orbital force: v 2 mv 2 F = m = r r The gravitational force that the object being orbited exerts on its satellite is GmM F= 2 r
- 12. Objects that are in orbit stay in orbit because the force required to keep them there is equal to the gravitational force that the object being orbited exerts on its satellite. If we set our two equations equal to each other and solve for v, we end up with a formula that will give us the orbital speed of the satellite.
- 13. Simplify the equation and solve for v: mv 2 GmM = 2 r r
- 14. Simplify the equation and solve for v: mv 2 GmM = 2 r r GmM 2 mv = r Gm v = r GM v= r 2
- 15. Because the mass of the satellite m cancelled out of the equation, if we know the orbital velocity and the radius of the orbit, we can find the mass of the object being orbited.
- 16. Rewrite the velocity equation and solve for M: GM v = r 2
- 17. Rewrite the velocity equation and solve for M: GM v = r 2 v 2r = GM v 2r M= G
- 18. Example: Use the Moon to calculate the mass of the Earth. Orbital radius: r = 3.84 × 10 8 m Period: T = 27.3 days circumference of orbit Orbital velocity: v = period of orbit
- 19. Example: Use the Moon to calculate the mass of the Earth. 2πr v= T = 2π ( 3.84 × 108 ) 24 hours 3600 seconds 27.3 1 hour 1 day = 1023 m s
- 20. Example: Use the Moon to calculate the mass of the Earth. G = 6.67 × 10 −11 N m2 kg2 v 2r M= G = 6.02 × 10 24 kg
- 21. To calculate escape velocity, we set the equation for kinetic energy to the equation for gravitational force and solve for v: Kinetic energy > Force × distance 1 GmM 2 mv > 2 i r 2 r 2GM 2 v > r v> 2GM r
- 22. Calculate Earth’s escape velocity in km/s. Earth’s mass: Earth’s radius: 6.02 × 1024 kg 6.38 × 106 m v > 11.22 km s
- 23. Now that we’ve worked through the different equations, we can calculate the mass and escape velocity of Mars as well as the mass of the Sun.
- 24. One of my favorite sites for possible astronomy-related math problems has been Space Math at http://spacemath.gsfc.nasa.gov. Unfortunately, because of cutbacks in NASA’s education budget, it will not be updated as frequently.
- 25. Invert the problem Ask for an explanation: oral or written Examples or counterexamples Ask for prediction Break into multiple parts Original (Standard) Problem Ask for multiple representation Ask for generalization Automaticity practice Ask questions that require qualitative reasoning James Epperson, Ph.D.
- 26. The powerpoint and the worksheets will be posted on my blog at tothemathlimit.wordpress.com.