2. A rocket, with initial mass 1500 kg , including 600 kg of fuel, is launched vertically upwards from rest. The rocket burns fuel at a rate of \(15 \mathrm {~kg} \mathrm {~s} ^ { - 1 }\) and the burnt fuel is ejected vertically downwards with a speed of \(1000 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to the rocket. At time \(t\) seconds after launch \(( t \leqslant 40 )\) the rocket has mass \(m \mathrm {~kg}\) and velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } + \frac { 1000 } { m } \frac { \mathrm {~d} m } { \mathrm {~d} t } = - 9.8$$
2. Find \(v\) at time \(t , 0 \leqslant t \leqslant 40\)