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Chapter 2: Q97E (page 68)
Show thatfollows from expressions (2-22) and (2-24) for momentum and energy in terms of m and u.
The relation of relativistic energy in terms of mass and momentum is derived bysquaring the relativistic energy relation in terms of mass and velocity andexpanding it using the binomial identity.
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According to Bob on Earth, Planet Y (uninhabited) is 5 ly away. Anna is in a spaceship moving away from Earth at 0.8c. She is bound for planet Y to study its geology. Unfortunately, Planet Y explodes. According to Bob.This occurred 2 yr after Anna passed Earth. (Bob. of course. has, to wait a while for the light from the explosion to arrive, but he reaches his conclusion by 鈥渨orking backward鈥) Call the passing of Anna and Bob time zero for both. (a) According to Anna, how far away is Planet Y when it explodes? (b) At what time does it explode?
For the situation given in Exercise 22, find the Lorentz transformation matrix from Bob鈥檚 frame to Anna鈥檚 frame, then solve the problem via matrix multiplication.
Equation (2-30) is an approximation correct only if the gravitational time-dilation effect is small. In this exercise, it is also assumed to be small. but we still allow for a nonuniform gravitational field. We start with (2-29), based on the Doppler effect in the accelerating frame. Consider two elevations, the lower at r1 and the upper at . Equation (2路29) becomes
Similarly, if we consider elevationsdata-custom-editor="chemistry" and data-custom-editor="chemistry" , we have
We continue the process, incrementing r by dr, until we reach r2.
Now imagine multiplying the left sides of all the equations and setting the product equal to the product of all the right sides. (a) Argue that the left side of the product is simply . (b) Assuming that the term in each individual equation is very small. so that productsof such termscan be ignored, argue that the right side of the product is
(c) Deduce from Newton鈥檚 universal law of gravitation, then argue that equation (2-31) follows from the result, just as (2-30) does from (2-29).
Bob is watching Anna fly by in her new high-speed plane, which Anna knows to be in length. As a greeting, Anna turns on two lights simultaneously, one at the front and one at the tail. According to Bob, the lights come apart.
(a) Which comes on first?
(b) How fast is the plane moving?
Question: Show that equation (2-36) follows from the arbitrary four-vector Lorentz transformation equations (2-35).
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