3. The engine of a car of mass 800 kg works at a constant rate of 32 kW . The car travels along a straight horizontal road and the resistance to motion of the car is proportional to the speed of the car. The car starts from rest and \(t\) seconds later it has a speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that $$800 v \frac { \mathrm {~d} v } { \mathrm {~d} t } = 32000 - k v ^ { 2 } , \text { where } k \text { is a positive constant. }$$ Given that the limiting speed of the car is \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find
2. the value of \(k\),
3. \(v\) in terms of \(t\).