The Relativity of Velocities — Question 8

Question 8

Rapidity makes repeated collinear velocity changes additive

Define rapidity $\eta=\operatorname{artanh}(v/c)$. Three successive frames each move at $+0.50c$ relative to the preceding frame. Use $\tanh(a+b)=(\tanh a+\tanh b)/(1+\tanh a\tanh b)$.

1. Find the final speed relative to the first frame.

2. Find its Lorentz factor and compare with the classical total.

Question 8 - Solution

Rapidity makes repeated collinear velocity changes additive

(a) Method and calculation

$$\eta_{\mathrm{total}}=3\operatorname{artanh}(1/2).$$$$\boxed{\beta_{\mathrm{total}}=\frac{3(1/2)+(1/2)^3}{1+3(1/2)^2}=13/14}.$$

(b) Interpretation and further calculation

$$\boxed{\gamma_{\mathrm{total}}=\frac1{\sqrt{1-(13/14)^2}}=\frac{14}{3\sqrt3}}.$$$$\beta_{\mathrm{classical}}=1.50,\qquad\boxed{\beta_{\mathrm{total}}\simeq0.928571<1}.$$

Physical check

Rapidities add for collinear boosts; velocities themselves do not.