8 A stone, of mass \(m\), is moving in a straight line along smooth horizontal ground.
At time \(t\), the stone has speed \(v\). As the stone moves, it experiences a total resistance force of magnitude \(\lambda m v ^ { \frac { 3 } { 2 } }\), where \(\lambda\) is a constant. No other horizontal force acts on the stone.
- Show that
$$\frac { \mathrm { d } v } { \mathrm {~d} t } = - \lambda v ^ { \frac { 3 } { 2 } }$$
(2 marks)
- The initial speed of the stone is \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Show that
$$v = \frac { 36 } { ( 2 + 3 \lambda t ) ^ { 2 } }$$
(7 marks)
- Find, in terms of \(\lambda\), the time taken for the speed of the stone to drop to \(4 \mathrm {~ms} ^ { - 1 }\).