Question

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?

Short answer

(a) The velocity of the speed of Carl relative to Bob’s spaceship is 0.357c .

(b) According, the speed of the Carl relative to Earth is 1.157c.

Step-by-step solution

Write the given data from the question

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

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

Determine the formulas to calculate the speed of the Carl relative Ann’s spaceship and speed of Carl relative to Earth.

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 the projectile relative Anna’s spaceship

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 .

Calculate the speed of the speed of the Carl relative to Earth.

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.