Answer
a. 0.96c.
b. 0.36c.
Work Step by Step
a. Let the direction of the first spaceship be the positive direction.
Reference frame S is the Earth and reference frame is the first spaceship. We see that v = 0.65c. The speed of the second spaceship relative to the first spaceship is given as u’=0.82c. Use equation 26–10 to solve for the speed of the second spaceship relative to the Earth.
$$u=\frac{v+u’}{(1+\frac{vu’}{c^2})}=\frac{0.65c+0.82c}{1+(0.65)(0.82)}=0.96c$$
b. Now, $u’=-0.82c$.
$$u=\frac{v+u’}{(1+\frac{vu’}{c^2})}=\frac{0.65c-0.82c}{1+(0.65)(-0.82)}=-0.36c$$
The problems asks for "speed", so the answer is 0.36c.