Answer
Relative to us, ship 3 has the largest speed, followed by a tie of ship 1 and ship 2, then ship 4.
$3\gt1=2\gt4$
Work Step by Step
We are given that The speeds $v_c$ of the cruisers relative to you and the speeds $v_s$ of the shuttle craft relative to their respective starships are, in that order, (1) 0.70c, 0.40c; (2) 0.40c, 0.70c; (3) 0.20c, 0.90c; (4) 0.50c, 0.60c.
To rank them according to their speeds relative to us, greatest first, we can look at equation 37-29, the relativity of velocities.
$$u=\frac{u^{'}+v}{1+u^{'}v/c^2}$$
Where $u$ is the velocity relative to us, $u^{'}$ is the rescue ship's velocity relative to the cruiser, and $v$ is the cruiser's speed.
$u^{'}+v$ is 1.1c in every case. What changes is $u^{'}+v$ in the denominator. The larger it is, the slower the ship will be moving relative to us.
Ship 1) $0.70c* 0.40c=2.8c^2$
Ship 2) $0.40c*0.70c=2.8c^2$
Ship 3) $0.20c* 0.90c=1.8c^2$
Ship 4) $0.50c* 0.60c=3.0c^2$
Relative to us, ship 3 has the largest speed, followed by a tie of ship 1 and ship 2, then ship 4.
