Question

A rocket fires a projectile at a speed of 0.95c while traveling past the earth. An earthbound scientist measures the projectile’s speed to be 0.90c. What was the rocket’s speed as a fraction of c?

Short answer

The speed of rocket is 0.34 c.

Step-by-step solution

Given information

We have given that, a rocket fires a projectile at a speed of 0.95c while traveling past the earth. An earthbound scientist measures the projectile’s speed to be 0.90c.

Step 1.

Let the earth’s frame of reference is S and the rocket’s frame of reference is S ′. Speed of projectile while passing earth u = 0.95 c and speed of projectile observed by scientist in earth's frame of reference u' = 0.90 c.

Step 2. 

Using Lorentz velocity transformation equation to calculate the speed of rocket :

$v=\frac{u-u\text{'}}{1-{\displaystyle \raisebox{1ex}{$uu\text{'}$}\!\left/ \!\raisebox{-1ex}{$c^2$}\right.}}$

$v=\frac{0.90c-0.95c}{1-{\displaystyle \raisebox{1ex}{$(0.90c)(0.95c)$}\!\left/ \!\raisebox{-1ex}{$c^2$}\right.}}$

$v=0.34c.$