Meriam, Kraige & Bolton — Kinetics of Particles: Problem 3_278

3/278 One of the functions of the space shuttle is to release communications satellites at low altitude. A booster rocket is fired at $B$, placing the satellite in an elliptical transfer orbit, the apogee of which is at the altitude necessary for a geosynchronous orbit. (A geosynchronous orbit is an equatorial-plane circular orbit whose period is equal to the absolute rotational period of the earth. A satellite in such an orbit appears to remain stationary to an earth-fixed observer.) A second booster rocket is then fired at $C$, and the final circular orbit is achieved. On one of the early space shuttle missions, a 1500-lb satellite was released from the shuttle at $B$, where $h_1 = 170$ miles. The booster rocket was to fire for $t = 90$ seconds, forming a transfer orbit with $h_2 = 22,300$ miles. The rocket failed during its burn. Radar observations determined the apogee altitude of the transfer orbit to be only 700 miles. Determine the actual time $t'$ which the rocket motor operated before failure. Assume negligible mass change during the booster rocket firing.