Answer
The magnitude of acceleration is $4.5~m/s^2$
Work Step by Step
To barely avoid a collision, the Enterprise needs to decelerate to a speed of 20 km/s while gaining 100 km on the other ship. The average speed during the deceleration period is 35 km/s, so the relative speed between the two ships is 15 km/s on average. We can find the time $t$ it takes to gain 100 km on the other ship;
$t = \frac{100~km}{15~km/s} = 6.67~s$
We can then find the rate of the deceleration:
$a = \frac{v-v_0}{t} = \frac{20~km/s-50~km/s}{6.67~s}$
$a = -4.5~km/s^2$
The magnitude of acceleration is thus $4.5~km/s^2=4500 m/s^2$.
