7. A rocket is fired from a fixed point \(O\). During the first phase of its motion its velocity, \(v \mathrm { ms } ^ { - 1 }\), is given at time \(t\) seconds after firing by the formula $$v = p t ^ { 2 } + q t .$$ 5 seconds after firing, the rocket is travelling at \(500 \mathrm {~ms} ^ { - 1 }\).
30 seconds after firing, the rocket is travelling at \(12000 \mathrm {~ms} ^ { - 1 }\).

1. Find the constants \(p\) and \(q\).
2. Sketch a velocity-time graph for the rocket for \(0 \leq t \leq 30\).
3. Find the initial acceleration of the rocket.
4. Find the distance of the rocket from \(O 30\) seconds after firing. From time \(t = 30\) onwards, the rocket maintains a constant speed of \(12000 \mathrm {~ms} ^ { - 1 }\).
5. Find the average speed of the rocket during its first 50 seconds of motion.