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GRE Subject Test: Physics : Four Vectors

Study concepts, example questions & explanations for GRE Subject Test: Physics

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Example Question #1 : Four Vectors

A reference frame S' is moving at 0.6c 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?

Possible Answers:

(1, 0, 2, 3)

(1, 0, 2.64, 3.3)

(1, 0, 3.3, 2.64)

(1, 0, 5.25, 4.75)

(1, 0, 4.75, 5.25)

Correct answer:

(1, 0, 4.75, 5.25)

Explanation:

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

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

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

Then, apply the Lorentz transformation:

x=x'=1 $\displaystyle x=x'=1$

y=y'=0 $\displaystyle y=y'=0$

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

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

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Example Question #2 : Four Vectors

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?

$n_{water}=1.33$ $\displaystyle n_{water}=1.33$

Possible Answers:

3 c $\displaystyle 3 c$

$\frac{9}{29}c$ $\displaystyle \frac{9}{29}c$

$\frac{27}{29}c$ $\displaystyle \frac{27}{29}c$

$\frac{27}{47}c$ $\displaystyle \frac{27}{47}c$

$\frac{9}{47}c$ $\displaystyle \frac{9}{47}c$

Correct answer:

$\frac{27}{29}c$ $\displaystyle \frac{27}{29}c$

Explanation:

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

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

To find the relative speed, use the following equation:

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

$v\approx0.6c$ $\displaystyle v\approx0.6c$

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GRE Subject Test: Physics : Four Vectors

Study concepts, example questions & explanations for GRE Subject Test: Physics

All GRE Subject Test: Physics Resources

Example Questions

Example Question #1 : Four Vectors

Example Question #2 : Four Vectors

All GRE Subject Test: Physics Resources

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