Question

2. A cube has a (proper) volume of 1000 cm cube.
Find the volume as determined by an observer O' who
moves at velocity of 0.8c relative to the cube in a
direction parallel to the length.​

Answer 1

The volume as determined by an observer O' who  moves at velocity of 0.8c relative to the cube in a  direction parallel to the length is 600 cm³

Explanation:

Volume of cube = 1000 cm³

If the side of the cube is L then

$L^3=1000$

$\implies L=\sqrt[3]{1000}=10$ cm

Velocity of the observer v = 0.8c

When the observer moves, the length contraction is given by

$\boxed{L'=L\sqrt{1-\frac{v^2}{c^2}}}$

$\implies L'=10\sqrt{1-\frac{(0.8c)^2}{c^2}}$

$\implies L'=10\sqrt{1-0.64}$

$\implies L'=10\sqrt{0.36}$

$\implies L'=10\times 0.6$

$\implies L'=6$ cm

There will be no contraction in breadth and height of the cube

Thus, the volume of the cube relative to the observer O'

$=L'\times L\times L$

$=6\times 10\times 10$

$=600$ cm³

Hope this answer is helpful.

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