Question

Two spaceships leave the Earth at time t-0 in opposite directions with velocities v1 and v2 (V1 > v2). At a subsequent time ts two light signals are sent from the Earth to both of the spaceships. E, and E2 are the events of the light signal reached at spaceship 1 and spaceship 2 respectively. (a) Find ty and xt of E, and the spatial coordinates of E, and E2 as viewed from (30 marks) (b) Obtain the ratio between vi and v2 in order that the reception of the signals (30 marks) (c) Draw the spacetime diagram of E1 and E2 in the reference frame of the Earth (40 marks) the Earth. can be simultaneous in the reference frame fixed to the faster spaceship. and spaceship 1. Mark all the relevant events and the worldlines.

Answer 1

(a)

I'll write everything in earth frame.

The light catches up to ship 1 at $t_1$, then

$x_1=v_1t_1=c(t_1-t_s)$

so that

$t_1=\frac{t_s}{1-v_1/c},\quad x_1=\frac{v_1t_s}{1-v_1/c}$

and similarly, for ship 2:-

$t_2=\frac{t_s}{1-v_2/c},\quad x_2=-\frac{v_2t_s}{1-v_2/c}$

(b)We have both t's and x's in earth frame. Transform them into ship 1's frame:

$t_1'=\gamma_1(t_1-v_1x_1/c^2)$

where

$\gamma_1=\frac{1}{\sqrt{1-v_1^2/c^2}}$

similarly,

$t_2'=\gamma_1(t_2-v_1x_2/c^2)$

according to the requirement

$t_2'=t_1'$

$t_1(1-v_1^2/c^2)=t_2(1+v_1v_2/c^2)$

using expressions of t's

$(1-v_2/c)(1-v_1^2/c^2)=(1-v_1/c)(1+v_1v_2/c^2)$

assuming v₁ is not 1

$(1-v_2/c)(1+v_1/c)=(1+v_1v_2/c^2)$

$2v_1v_2=(v_1-v_2)c$

(c)

In earth frame

1 1 2 1 2 2

dashed lines represent 45 degree lines, in c=1 units.

In Ship 1 frame

where red is the earth's world line and

$t'_s=\gamma_1t_s$

and the world line of ship 1 coincides with time axis.