Acceleration as function of position

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.

1. Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = - \lambda v ^ { \frac { 3 } { 2 } }$$ (2 marks)
2. 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)
3. Find, in terms of \(\lambda\), the time taken for the speed of the stone to drop to \(4 \mathrm {~ms} ^ { - 1 }\).

17 A ball is released from a great height so that it falls vertically downwards towards the surface of the Earth. 17

1. Using a simple model, Andy predicts that the velocity of the ball, exactly 2 seconds after being released from rest, is \(2 g \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Show how Andy has obtained his prediction.
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2. Using a refined model, Amy predicts that the ball's acceleration, \(a \mathrm {~ms} ^ { - 2 }\), at time \(t\) seconds after being released from rest is $$a = g - 0.1 v$$ where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of the ball at time \(t\) seconds. Find an expression for \(v\) in terms of \(t\).
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3. Comment on the value of \(v\) for the two models as \(t\) becomes large.