Four Vectors - GRE Subject Test: Physics

Card 0 of 2

Question

A container of water is carried on a space ship traveling at 60% the speed of light relative to the Earth. From the perspective of an observer on Earth, what is the speed of the light in the water?

$n_{water}=1.33$

Answer

The speed of light through water in a stationary frame is found from the index of refraction of water, which is approximately $\frac{4}{3}$

$v'=\frac{c}{n}=\frac{3}{4}c$

To find the relative speed, use the following equation:

$v_{p}=\frac{v'+v}{1+\frac{vv'}{c^2}}$

$v\approx0.6c$

Compare your answer with the correct one above

Question

A reference frame S' is moving at 0.6_c_ in the z direction with respect to a stationary frame S. An event occurs in S' with the coordinates (x', y', z', ct')=(1, 0, 2, 3). What are the coordinates (x, y, z, ct) of the event with respect to the stationary frame S?

Answer

To apply a Lorentz transformation, we need gamma and beta:

$\beta=\frac{v}{c}=\frac{3}{5}$

$\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}=\frac{5}{4}$

Then, apply the Lorentz transformation:

$z=\gamma(z'+\beta ct')=\frac{5}{4}(2+\frac{3}{5}\cdot 3)=4.75$

$ct=\gamma(ct'+\beta z')=\frac{5}{4}(3+\frac{3}{5}\cdot 2)=5.25$

Compare your answer with the correct one above

Tap the card to reveal the answer