Homework 7

5/20/02

Physics 110B

Griffiths 12.26, 12.27, 12.29, 12.30, 12.34

1. Consider two points traveling at the speed of light and separated by a distance $\Delta$, in an inertial frame $S$. We can write $x_1 = ct$ and $x_2 = ct + \Delta$ . In this problem you will find out how these points are described in a reference frame $S'$ traveling to the right at a speed of $v$ with respect to $S$. The easiest way to do this is to use the Lorentz transformation which gives $(t,x)$ in terms of $(t',x')$.

(a)

Write down the above Lorentz transformation: $x = function(t',x')$ and $t = function(t',x')$.

(b)

What is the equation relating the position of the first point $x'_1$, to $t'$? Do this by writing $x_1 = ct$ and plugging in the results from $(a)$. Your answer should be simple and make a lot of sense!

(c)

What is the equation relating the position of the second point $x'_2$, to $t'$?

(d)

What is the separation between the points $x'_2 - x'_1$?

Why should anyone be interested? If you interpret points $x_1$ and $x_2$ as the nodes of an electro-magnetic plane wave, then the difference is the wavelength. This is therefore a calculation of the Doppler shift for light.

2. You are a TV hosts for the tenth annual ``Nerdathon''. Two contestants push their answer buttons almost simultaneously. (The question concerns the color of Einstein's socks when traveling at half the speed of light, see previous problem for the answer). Because you are happy, you are skipping between the contestants at an approximate speed of $v$. According to the audience, the interval between the two buzzes is spacelike. In other words the audience says that the difference in position between the contestants $\Delta x$ divided by the time between the buzzes $\Delta t$ is greater than the velocity of light.

(a) Will the interval appear spacelike or timelike to you?

(b) At what speed $v$ would you have to be skipping in order for the two buzzes to appear simultaneous to you?

3. During the Nerdathon you somehow got lost. You find yourself standing on some railroad track between two high speed trains that are going towards you with equal and opposite velocities. The engineer on one train says his speed is $v$ relative to the other train. What is the train's speed relative to you?

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