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Chapter 35: Problem 18

Prove that in all cases, two sub-light-speed velocities "added" relativistically will always yield a sub-light-speed velocity. Consider motion in one spatial dimension only.

Short Answer

Expert verified
Question: Prove that the total sub-light-speed velocity obtained from adding two sub-light-speed velocities relativistically is always less than the speed of light. Answer: By using the relativistic velocity addition formula and analyzing the denominators, we demonstrated that the total velocity w is always sub-light-speed, meaning it is always less than the speed of light.

Step by step solution

01

Review the relativistic velocity addition formula

The relativistic addition of velocities formula is given as: $$ w = \frac{v + u}{1 + \frac{v \cdot u}{c^2}} $$ where w is the total velocity, v and u are the two sub-light-speed velocities being added, and c is the speed of light.
02

Define the given conditions

In our given problem, the velocities are sub-light-speed, which means \(0 \leq v < c\) and \(0 \leq u < c\). Our goal is to prove that the total velocity w is also sub-light-speed, so \(0 \leq w < c\).
03

Analyze the denominator of the formula

Let's analyze the denominator of the fraction \(\frac{v \cdot u}{c^2}\). Since both v and u are sub-light-speed, their product is positive and smaller than \(c^2\). Therefore, the fraction is positive and smaller than 1. $$ 0 < \frac{v \cdot u}{c^2} < 1 $$ Adding 1 to this inequality, $$ 1 < 1 + \frac{v \cdot u}{c^2} < 2 $$
04

Show that the total velocity is sub-light-speed

Now, we'll use the formula for relativistic velocity addition to prove that the total velocity w is sub-light-speed: $$ w = \frac{v + u}{1 + \frac{v \cdot u}{c^2}} $$ In the worst-case scenario (maximum), v and u are both close to c. Thus, \((v + u)\) is close to \(2c\), and the denominator from step 3 is less than 2. Then, the maximum value of w can be obtained when the numerator is the largest and the denominator is the smallest: $$ w_{max} = \frac{2c}{1 + \frac{v \cdot u}{c^2}} $$ Since the denominator is always greater than 1, it follows that the fraction is less than 2, so \(w_{max} < 2c\). Therefore, the total velocity w is always sub-light-speed, as required to prove.

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Most popular questions from this chapter

A meteor made of pure kryptonite (Yes, we know: There really isn't such a thing as kryptonite ...) is moving toward Earth. If the meteor eventually hits Earth, the impact will cause severe damage, threatening life as we know it. If a laser hits the meteor with wavelength \(560 \mathrm{nm}\), the entire meteor will blow up. The only laser powerful enough on Earth has a \(532-\mathrm{nm}\) wavelength. Scientists decide to launch the laser in a spacecraft and use special relativity to get the right wavelength. The meteor is moving very slowly, so there is no correction for relative velocities. At what speed does the spaceship need to move so the laser has the right wavelength, and should it travel toward or away from the meteor?

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