The Relativity of Velocities — Question 4

Question 4

A transverse launch acquires longitudinal motion and a reduced transverse component

A probe is launched with $u_x\prime=0$ and $u_y\prime=0.80c$ from a spacecraft moving at $v=0.60c$. 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 both laboratory velocity components and the total speed.

2. Find the total Lorentz factor and check it against the product of the spacecraft and launch factors.

Question 4 - Solution

A transverse launch acquires longitudinal motion and a reduced transverse component

(a) Method and calculation

$$\boxed{u_x=0.60c},\qquad\boxed{u_y=0.80c/1.25=0.64c}.$$$$\boxed{|\mathbf u|/c=\sqrt{.60^2+.64^2}=\sqrt{481}/25\simeq0.877268}.$$

(b) Interpretation and further calculation

$$\gamma_u=(1-481/625)^{-1/2}=\boxed{25/12}.$$$$\gamma_v\gamma_{u\prime}=(5/4)(5/3)=\boxed{25/12}.$$

Physical check

Transverse velocity does not stay unchanged because elapsed time transforms even when the transverse coordinate does not.