Question

Bob is on Earth. Anna is on a spacecraft moving away from Earth at 0.6c . At some point in Anna's outward travel, Bob fires a projectile loaded with supplies out to Anna's ship. Relative to Bob, the projectile moves at 0.8c . (a) How fast does the projectile move relative to Anna? (b) Bob also sends a light signal, " Greetings from Earth:' out to Anna's ship. How fast does the light signal move relative to Anna?

Short answer

(a) The velocity of the projectile relative to Anna’s speed is 0.385c .

(b) The speed of the light beam relative to Anna is C.

Step-by-step solution

Write the given data from the question.

The velocity of the projectile, u=0.8 c

The velocity of Anna’s spacecraft, v=0.6 c

Determine the formulas to calculate the speed of the projectile relative Ann’s spaceship and speed of the light beam relative to Anna’s spaceship.

The expression for the Lorentz velocity transformation is given as follows.

$\mathbf{u}\mathbf{\text{'}}\mathbf{=}\frac{\mathbf{u}\mathbf{-}\mathbf{v}}{\mathbf{1}\mathbf{-}{\displaystyle \frac{\mathbf{uv}}{{\mathbf{c}}^{\mathbf{2}}}}}$ …… (i)

Here, u' is the velocity of the object in frame S' , u is the velocity of object in frame S , v is the velocity of frame S' relative to s and cis the speed of the light.

Calculate the speed of the projectile relative Anna’s spaceship

Calculate the speed of projectile relative to Anna’s spaceship.

Substitute 0.8c for u and 0.6c for v into equation (i).

$$\begin{gathered} u\text{'}=\frac{0.8c-0.6c}{1-{\displaystyle \frac{\left(0.8c\right)\left(0.6c\right)}{c^2}}} \\ u\text{'}=\frac{0.2c}{1-0.48} \\ u\text{'}=\frac{0.2c}{0.52} \\ u\text{'}=0.384c \end{gathered}$$

Hence the velocity of the projectile relative to Anna’s speed is 0.385c .

Calculate the speed of the light beam relative to Anna’s spaceship.

The Anna is moving with speed of the spaceship but Anna is stable with respect to the spaceship. The inertial observer will always see the same speed of light. The beam of the light moves with speed of the light relative to Anna.

Hence the speed of the light beam relative to Anna is c .