According Anna, on Earth, Bob is on a spaceship moving at 0.8c toward Earth, and Carl, a little farther out. is on a spaceship moving at 0.9c toward Earth. (a) According to Bob, how fast and in what direction is Carl moving relative to himself (Bob)? (b) According to Bob, how fast is Carl moving relative to Earth?

The velocity of the Bob’s spaceship, v=0.8 c

The velocity of carl’s spacecraft, u=0.9 c

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 c is the speed of the light.

Calculate the speed of Carl relative to Bob’s spaceship.

Substitute 0.8c for V and for 0.9c for u into equation (i).

$$\begin{gathered} u\text{'}=\frac{0.9c-0.8c}{1-{\displaystyle \frac{\left(0.9c\right)\left(0.8c\right)}{c^2}}} \\ u\text{'}=\frac{0.1c}{1-0.72} \\ u\text{'}=\frac{0.1c}{0.28} \\ u\text{'}=0.357c \end{gathered}$$

Hence the velocity of the speed of Carl relative to Bob’s spaceship is 0.357c .

The velocity Carl with respect to Earth is given by,

u'=u-v …… (ii)

Here, u is the apparent velocity of Bob to Carl and v is the apparent velocity of the Earth with respect to Carl.

Substitute 0.357c for u and -0.8c for V into equation (ii).

u'=0.357c-(-0.8c)

u'=0.357c+0.8c

u'=1.157c

Hence according, the speed of the Carl relative to Earth is 1.157c.