The Relativity of Velocities — Question 1

Question 1

Forward launch speeds combine through a denominator as well as a sum

A spacecraft moves at $v=0.60c$ in the laboratory and launches a probe forward at $u\prime=0.80c$ in the spacecraft frame. For $S\prime$ moving at +v along x, use $u_x=(u_x\prime+v)/(1+vu_x\prime/c^2)$ and $u_y=u_y\prime/[\gamma_v(1+vu_x\prime/c^2)]$, with $\gamma_v=(1-v^2/c^2)^{-1/2}$.

1. Find the laboratory probe speed.

2. Invert the result to recover the launch speed and compare with the classical sum.

Question 1 - Solution

Forward launch speeds combine through a denominator as well as a sum

(a) Method and calculation

$$\frac uc=\frac{0.80+0.60}{1+0.60(0.80)}=\boxed{35/37\simeq0.945946}.$$$$u_{\mathrm{classical}}=1.40c>c.$$

(b) Interpretation and further calculation

$$\frac{u\prime}c=\frac{u/c-0.60}{1-0.60u/c}.$$$$\boxed{u\prime/c=\frac{35/37-3/5}{1-(3/5)(35/37)}=4/5}.$$

Physical check

Both forward and inverse transformations preserve the physical subluminal range.