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Chapter 33: Problem 36

Two spaceships are racing. The "slower" one passes Earth at \(0.70 c,\) and the "faster" one moves at \(0.40 c\) relative to the slower one. What's the faster ship's speed relative to Earth?

Short Answer

Expert verified
The faster spaceship's speed relative to Earth is approximately \(0.86c\) or 86% the speed of light

Step by step solution

01

Understand and write down the Einstein velocity addition formula

The Einstein velocity addition formula is given by \[ u = \frac{v + u'}{1 + \frac{vu'}{c^2}}, \] where \(v\) is the slower spaceship's velocity relative to Earth, \(u'\) is the faster spaceship's velocity relative to the slower one, \(c\) is the speed of light in vacuum, and \(u\) is what we want to find - the faster spaceship's velocity relative to Earth.
02

Substitute the known values into the formula

\[ u = \frac{0.70c + 0.40c}{1 + \frac{0.70c \cdot 0.40c}{c^2}}, \] Here, both velocities \(v\) and \(u'\) are given relative to the speed of light \(c\). Therefore, they are multiplied by \(c\) in the formula.
03

Simplify the formula

Simplify the equations by reducing the terms and solving for \(u\). The \(c\) terms in the numerator and denominator cancel out since they are in each of the terms of the fraction, leaving: \[ u = \frac{0.70 + 0.40}{1 + (0.70 \cdot 0.40)} \]
04

Calculate the value of \(u\)

Insert the values and carry out the addition and multiplication: \[ u = \frac{1.10}{1 + 0.28} = \frac{1.10}{1.28} \] Calculating the above expression gives \(u 鈮� 0.86\) or \(86\%\) the speed of light.

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