QuestionJune 19, 2025

Question 7 (10 points) A relativistic train is moving at 0.6 c relative to the Earth A rocket of proper length 50m is observed to be 30m long by the observers in the train , as the rocket rushes past the train moving parallel to the train and in the same direction. What is the rocket's speed relative to Earth? 0.93 c 0.80 c 1.33 c None of the other answers 0.60 c 0.50 c

Question 7 (10 points) A relativistic train is moving at 0.6 c relative to the Earth A rocket of proper length 50m is observed to be 30m long by the observers in the train , as the rocket rushes past the train moving parallel to the train and in the same direction. What is the rocket's speed relative to Earth? 0.93 c 0.80 c 1.33 c None of the other answers 0.60 c 0.50 c
Question 7 (10 points) A relativistic train is moving at 0.6 c relative to the Earth A rocket of proper length 50m is observed to be 30m long by the observers in the train , as the rocket rushes past the train moving parallel to the train and in the same direction. What is the rocket's speed relative to Earth? 0.93 c 0.80 c 1.33 c None of the other answers 0.60 c 0.50 c

Solution
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Answer

0.80 c Explanation 1. Use Length Contraction Formula The formula for length contraction is L = L_0 \sqrt{1 - \frac{v^2}{c^2}}, where L is the observed length, L_0 is the proper length, and v is the velocity of the object relative to the observer. Rearrange to solve for v: v = c \sqrt{1 - \left(\frac{L}{L_0}\right)^2}. 2. Substitute Values Substitute L = 30\,m, L_0 = 50\,m, and c = 1 (speed of light in units of c) into the formula: v = c \sqrt{1 - \left(\frac{30}{50}\right)^2}. 3. Calculate Rocket's Speed Calculate v = c \sqrt{1 - \left(\frac{3}{5}\right)^2} = c \sqrt{1 - \frac{9}{25}} = c \sqrt{\frac{16}{25}} = c \cdot \frac{4}{5} = 0.8c.

Explanation

1. Use Length Contraction Formula<br /> The formula for length contraction is $L = L_0 \sqrt{1 - \frac{v^2}{c^2}}$, where $L$ is the observed length, $L_0$ is the proper length, and $v$ is the velocity of the object relative to the observer. Rearrange to solve for $v$: $v = c \sqrt{1 - \left(\frac{L}{L_0}\right)^2}$.<br /><br />2. Substitute Values<br /> Substitute $L = 30\,m$, $L_0 = 50\,m$, and $c = 1$ (speed of light in units of $c$) into the formula: $v = c \sqrt{1 - \left(\frac{30}{50}\right)^2}$.<br /><br />3. Calculate Rocket's Speed<br /> Calculate $v = c \sqrt{1 - \left(\frac{3}{5}\right)^2} = c \sqrt{1 - \frac{9}{25}} = c \sqrt{\frac{16}{25}} = c \cdot \frac{4}{5} = 0.8c$.
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