Question

A pursuit spacecraft from the planet Tatooine is attempting to catch up with a Trade Federation cruiser. As measured by an observer on Tatooine, the cruiser is traveling away from the planet with a speed of 0.600c. The pursuit ship is travelling at a speed of 0.800crelative to Tatooine, in the same direction as the cruiser. What is the speed of the cruiser relative to the pursuit ship?

Short answer

The speed of the cruiser relative to the pursuit ship is -0.385c.

Step-by-step solution

Velocity transformation in relativity

When two particle or a body moves with a velocity which is approaching to speed of light then normal method of finding relative velocity does not work as it exceeds the velocity of body greater than velocity of light.

Therefore relative velocity for two particles moving with velocity approaching to speed of light is given by velocity addition method.

Velocity addition method

According the velocity addition method the velocity of particle in lab frame observed by observer in moving frame is given by

$u_x^{\text{'}}=\frac{u_x-v}{1-{\displaystyle \frac{v{u}_x}{c^2}}}---\left(i\right)$

For the velocity of particle moving in moving frame or relativistic frame observed by observer in rest frame is given by inverse Lorentz transformation of first (i) formula

That is

$u_x=\frac{u_x^{\text{'}}+v}{c^2}$

Where, $u_x^{\text{'}}$ is the velocity of body in $S\text{'}$frame (moving frame), $u_x$is velocity of body in frame S (lab frame) and v is velocity of the moving frame.

The calculation of the speed of the cruiser relative to the pursuit ship

Given: Velocity of cruiser relative to planet $u_x=0.600c$,

Velocity of moving frame is same as the velocity of pursuit ship So, $v=0.800c$.

Using

$u_x^{\text{'}}=\frac{u_x-v}{1-{\displaystyle \frac{v{u}_x^{\text{'}}}{c^2}}}$

Now put the values of constants in above equation

$$\begin{gathered} u_x^{\text{'}}=\frac{0.600c-0.800c}{1-{\displaystyle \frac{\left(0.800c\right)\left(0.600c\right)}{c^2}}} \\ {u}_x^{\text{'}}=-0.385c \end{gathered}$$

Thus, the speed of the cruiser relative to pursuit ship is -0.385c