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Chapter 26: Problem 31

As observed from Earth, rocket Alpha moves with speed \(0.90 c\) and rocket Bravo travels with a speed of \(0.60 c\) They are moving along the same line toward a head-on collision. What is the speed of rocket Alpha as measured from rocket Bravo? (tutorial: adding velocities)

Short Answer

Expert verified
Answer: The relative velocity of rocket Alpha as seen from rocket Bravo is approximately 0.65217c or 65.217% of the speed of light.

Step by step solution

01

Identify the given information

We are given the velocities of rocket Alpha and rocket Bravo as observed from Earth or in the Earth's frame. - Velocity of rocket Alpha, \(v_A = 0.90c\) - Velocity of rocket Bravo, \(v_B = 0.60c\) We want to find the velocity of rocket Alpha as measured from rocket Bravo.
02

Apply the Einstein velocity addition formula

The Einstein velocity addition formula is used to add velocities in Special Relativity. The formula is given by: \(u = \frac{v_A + v_B}{1 + \frac{v_A v_B}{c^2}}\) where \(u\) is the relative velocity of rocket Alpha as measured from rocket Bravo, \(v_A\) is the velocity of rocket Alpha, \(v_B\) is the velocity of rocket Bravo, and \(c\) is the speed of light.
03

Insert the given values into the formula and solve

Plugging the given values into the formula, we get: \(u = \frac{0.90c - 0.60c}{1 + \frac{(0.90c)(-0.60c)}{c^2}}\) Notice that for rocket Bravo, its speed is in the opposite direction, so its value becomes negative. Simplify the expression and solve for \(u\): \(u = \frac{0.30c}{1 - 0.54}\) \(u = \frac{0.30c}{0.46}\) \(u = 0.65217c\)
04

Report the final answer

The speed of rocket Alpha as measured from rocket Bravo is approximately \(0.65217c\) or \(65.217\%\) of the speed of light.

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