The Relativity of Velocities — Question 2

Question 2

Opposing beams have a relative speed smaller than their closing rate

In one laboratory, beam A moves at $+0.80c$ and beam B at $-0.60c$.

1. Use $u_B\prime=(u_B-v_A)/(1-u_Bv_A/c^2)$ to find B velocity in A rest frame.

2. Compare the relative speed with the laboratory closing rate and identify what each measures.

Question 2 - Solution

Opposing beams have a relative speed smaller than their closing rate

(a) Method and calculation

$$\frac{u_B\prime}c=\frac{-0.60-0.80}{1-(-0.60)(0.80)}.$$$$\boxed{u_B\prime=-35c/37\simeq-0.945946c}.$$

(b) Interpretation and further calculation

$$-\frac{d(x_B-x_A)}{dt}=0.80c+0.60c=\boxed{1.40c}.$$$$\boxed{|u_B\prime|<c},\qquad\text{closing rate uses two laboratory positions}.$$

Physical check

Changing frames changes the time coordinate in the velocity ratio. Subtracting laboratory velocities alone does not perform that change.