- A raindrop falls from rest from a cloud. The velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically downwards, of the raindrop, \(t\) seconds after the raindrop starts to fall, is modelled by the differential equation
$$( t + 2 ) \frac { \mathrm { d } v } { \mathrm {~d} t } + 3 v = k ( t + 2 ) - 3 \quad t \geqslant 0$$
where \(k\) is a positive constant.
- Solve the differential equation to show that
$$v = \frac { k } { 4 } ( t + 2 ) - 1 + \frac { 4 ( 2 - k ) } { ( t + 2 ) ^ { 3 } }$$
Given that \(v = 4\) when \(t = 2\)
- determine, according to the model, the velocity of the raindrop 5 seconds after it starts to fall.
- Comment on the validity of the model for very large values of \(t\)