A rocket speeding by the Earth is roughly cylindrical in shape and oriented such that its axis is parallel to its velocity. At rest, it is measured to have a radius of 15 m and a length of 150 m. When speeding by, its volume is measured to be 4.24 × 10⁴ m³. What is its speed? A) 0.917 c B) 0.774 c C) 0.678 c D) Insufficient information is given.

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Quizplus · Accepted Answer

The volume of the rocket contracts due to Lorentz contraction in the direction of motion, but the radius (perpendicular to the direction of motion) remains unchanged. The original volume is $V_0 = \pi r^2 l = \pi (15)^2 (150)$. The contracted volume is $V = 4.24 \times 10^{14}$. Since only the length contracts, $V = \pi r^2 l'$, where $l'$ is the contracted length. The ratio of the volumes gives the Lorentz contraction factor: $\frac{V}{V_0} = \frac{l'}{l} = \sqrt{1 - v^2/c^2}$. Solving for $v$, we find $v \approx 0.917c$.

A Rocket Speeding by the Earth Is Roughly Cylindrical in Shape