A particle of mass m is sent at a speed u towards another mass m particle at rest. Afterwards the two mass m particles are observed to fly off symmetrically, as shown (this isn't the only option for such a collision, of course, but it is one option). In conference section you showed that a non-relativistic analysis leads to the conclusion that the final velocities are at right angles to one another, i.e. 0 = 45°. But, now we have expressions for relativistic energy and momentum, so you should be able to solve the relativistic problem to find the angle 0. Before After (a) Use relativistic energy and momentum conservation to find 0. To get started, write out conditions for conservation of energy and momentum in the “lab frame" (as pictured). You should be able to make substitutions to write all of your momenta in terms of the energy (call it E1) of the initial incoming particle and derive an expression for cos 0 which depends only on E1,m,, and c. This is much like the collision problem from Conference Section 2, except now we are looking at a 2D collision. (b) Show that 0 depends only on yu, not m. (c) Take the limit of cos 0 when u/c « 1. This low-speed limit of the relativistic expression should agree with the non-relativisitc version (0 = 45°). Does it? %3D
A particle of mass m is sent at a speed u towards another mass m particle at rest. Afterwards the two mass m particles are observed to fly off symmetrically, as shown (this isn't the only option for such a collision, of course, but it is one option). In conference section you showed that a non-relativistic analysis leads to the conclusion that the final velocities are at right angles to one another, i.e. 0 = 45°. But, now we have expressions for relativistic energy and momentum, so you should be able to solve the relativistic problem to find the angle 0. Before After (a) Use relativistic energy and momentum conservation to find 0. To get started, write out conditions for conservation of energy and momentum in the “lab frame" (as pictured). You should be able to make substitutions to write all of your momenta in terms of the energy (call it E1) of the initial incoming particle and derive an expression for cos 0 which depends only on E1,m,, and c. This is much like the collision problem from Conference Section 2, except now we are looking at a 2D collision. (b) Show that 0 depends only on yu, not m. (c) Take the limit of cos 0 when u/c « 1. This low-speed limit of the relativistic expression should agree with the non-relativisitc version (0 = 45°). Does it? %3D
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
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Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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Please make sure to use relativistic energy and momentum and not Newtonian. Thanks!
Transcribed Image Text:8) Equal Mass Collision
A particle of mass m is sent at a speed u towards another mass m particle at rest. Afterwards the two
mass m particles are observed to fly off symmetrically, as shown (this isn't the only option for such
a collision, of course, but it is one option). In conference section you showed that a non-relativistic
analysis leads to the conclusion that the final velocities are at right angles to one another, i.e. 0 = 45°.
But, now we have expressions for relativistic energy and momentum, so you should be able to solve
the relativistic problem to find the angle 0.
Before
After
(a) Use relativistic energy and momentum conservation to find 0.
To get started, write out conditions for conservation of energy and momentum in the "lab frame"
(as pictured). You should be able to make substitutions to write all of your momenta in terms
of the energy (call it E1) of the initial incoming particle and derive an expression for cos 0 which
depends only on E1, m,, and c. This is much like the collision problem from Conference Section 2,
except now we are looking at a 2D collision.
(b) Show that 0 depends only on yu, not m.
(c) Take the limit of cos 0 when u/c « 1. This low-speed limit of the relativistic expression should
agree with the non-relativisitc version (0 = 45°). Does it?
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