Answer
The astronaut has 1.63 seconds to fire the rockets.
Work Step by Step
We can find the orbital speed for an object that is 400 km above the earth's surface. Note that the distance from the center of the earth is 6380 km + 400 km which is 6780 km.
$v = \sqrt{\frac{G~M_e}{R}}$
$v = \sqrt{\frac{(6.67\times 10^{-11}~m^3/kg~s^2)(5.98\times 10^{24}~kg)}{6.78\times 10^6~m}}$
$v = 7.67\times 10^3~m/s$
$v = 7.67~km/s$
Since both objects are moving at a speed of 7.67 km/s but in opposite directions, the relative speed is 15.34 km/s. We can find the time it takes the two objects to cover a distance of 25 km.
$t = \frac{distance}{speed}$
$t = \frac{25~km}{15.34~km/s}$
$t = 1.63~s$
The astronaut has 1.63 seconds to fire the rockets.
