Before the engines fail, the rocket's altitude at time <em>t</em> is given by

and its velocity is

The rocket then reaches an altitude of 1150 m at time <em>t</em> such that

Solve for <em>t</em> to find this time to be

At this time, the rocket attains a velocity of

When it's in freefall, the rocket's altitude is given by

where  is the acceleration due to gravity, and its velocity is
 is the acceleration due to gravity, and its velocity is

(a) After the first 11.2 s of flight, the rocket is in the air for as long as it takes for  to reach 0:
 to reach 0:

So the rocket is in motion for a total of 11.2 s + 32.6 s = 43.4 s.
(b) Recall that

where  and
 and  denote final and initial velocities, respecitively,
 denote final and initial velocities, respecitively,  denotes acceleration, and
 denotes acceleration, and  the difference in altitudes over some time interval. At its maximum height, the rocket has zero velocity. After the engines fail, the rocket will keep moving upward for a little while before it starts to fall to the ground, which means
 the difference in altitudes over some time interval. At its maximum height, the rocket has zero velocity. After the engines fail, the rocket will keep moving upward for a little while before it starts to fall to the ground, which means  will contain the information we need to find the maximum height.
 will contain the information we need to find the maximum height.

Solve for  and we find that the rocket reaches a maximum altitude of about 1930 m.
 and we find that the rocket reaches a maximum altitude of about 1930 m.
(c) In part (a), we found the time it takes for the rocket to hit the ground (relative to  ) to be about 32.6 s. Plug this into
) to be about 32.6 s. Plug this into  to find the velocity before it crashes:
 to find the velocity before it crashes:

That is, the rocket has a velocity of 196 m/s in the downward direction as it hits the ground.