Savchenko Problem 14.1.10

Physics · mechanics · electromagnetism

Problem

a. Based on observations from Earth, the velocities of two spacecraft flying toward each other are equal to v and u. Show that the relative velocity of one spacecraft from observations from the other is determined by the formula

v = \frac{v+u}{1+\frac{vu}{c^2}}

. b. Based on observations from Earth, the spacecraft is moving away from it at a speed of v. A probe was ejected from the ship in the direction of travel. According to observations from the ship, the probe moves relative to the ship at a speed of u. Prove that the velocity of the probe's distance from the Earth observed from Earth is

\frac{v+u}{1+\frac{vu}{c^2}}

. When solving problems, use the constancy of the speed of light in different reference systems.

Intuitive explanation

This problem is about relativistic velocity addition—how velocities combine when they're comparable to the speed of light. The key insight is that Galilean addition ( $v + u$ ) fails at high speeds because the speed of light must be the same in all reference frames. The denominator $1 + \frac{vu}{c^2}$ is the relativistic correction that ensures no object ever exceeds $c$ , no matter how you add velocities. Both parts use the same formula, just applied to different observer pairs.

Intuitive explanation

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