Limiting or terminal velocity

4. A body falls vertically from rest and is subject to air resistance of a magnitude which is proportional to its speed. Given that its terminal speed is \(100 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the time it takes for the body to attain a speed of \(60 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
(10 marks)

2 A car of mass \(m \mathrm {~kg}\) starts from rest at a point O and moves along a straight horizontal road. The resultant force in the direction of motion has power \(P\) watts, given by \(P = m \left( k ^ { 2 } - v ^ { 2 } \right)\), where \(v \mathrm {~ms} ^ { - 1 }\) is the velocity of the car and \(k\) is a positive constant. The displacement from O in the direction of motion is \(x \mathrm {~m}\).

1. Show that \(\left( \frac { k ^ { 2 } } { k ^ { 2 } - v ^ { 2 } } - 1 \right) \frac { \mathrm { d } v } { \mathrm {~d} x } = 1\), and hence find \(x\) in terms of \(v\) and \(k\).
2. How far does the car travel before reaching \(90 \%\) of its terminal velocity?

The displacement of \(P\) from \(O\) is \(x\) m at time \(t\) s.

1. Find an expression for \(x\) in terms of \(t\), while \(P\) is moving upwards. [2]
2. Find, correct to 3 significant figures, the greatest height above \(O\) reached by \(P\). [2]

A particle \(P\) of mass \(5\) kg moves along a horizontal straight line. At time \(t\) s, the velocity of \(P\) is \(v\) m s\(^{-1}\) and its displacement from a fixed point \(O\) on the line is \(x\) m. The forces acting on \(P\) are a force of magnitude \(\frac{500}{v}\) N in the direction \(OP\) and a resistive force of magnitude \(\frac{1}{2}v^2\) N. When \(t = 0\), \(x = 0\) and \(v = 5\).

1. Find an expression for \(v\) in terms of \(x\). [6]
2. State the value that the speed approaches for large values of \(x\). [1]