Question

Suppose that Alice is moving to your right at speed c/2 (relative to you)
and Bob is moving to Alice’s right at speed c/2 relative to her.
(a) What is the boost parameter between you and Alice?
(b) Between Alice and Bob?
(c) Between you and Bob? Use this to calculate the relative velocity
between you and Bob. [Hint: Your calculator almost certainly
7 has
a built-in function called tanh−1
or arctanh. The relation between θ
and v can be written as θ = tanh−1
(v/c) = arctanh(v/c).]

Answer 1

Boost parameter between you and Alice

$$\theta_{YA}=arctanh(\frac{v_{YA}}{c})=0.549$$

Boost parameter between Bob and Alice

$$\theta_{BA}=arctanh(\frac{v_{BA}}{c})=0.549$$

Finally, boost parameter between You and Bob

$$\theta_{YB}=\theta_{BA}+\theta_{YA}=1.099$$

Relative velocity between You and Bob

$$v_{YB}/c=tanh(\theta_{YB})=tanh(1.099)=0.8$$ $$v_{YB}=0.8c$$