M4 June 2014 (R) Q3
3. A small ball of mass \(m\) is projected vertically upwards from a point \(O\) with speed \(U\). The ball is subject to air resistance of magnitude \(mkv\), where \(v\) is the speed of the ball and \(k\) is a positive constant.
Find, in terms of \(U\), \(g\) and \(k\), the maximum height above \(O\) reached by the ball. (8)
| Scheme | Marks |
|---|---|
| \(-(mg + mkv) = mv\dfrac{\mathrm{d}v}{\mathrm{d}x}\) Differential equation | M1 A1 |
| \(\displaystyle\int_0^H \mathrm{d}x = \int_U^0 -\frac{v}{(g + kv)}\,\mathrm{d}v\) Separate variables | M1 |
| \(\displaystyle\int_0^H \mathrm{d}x = -\int_U^0 \frac{1}{k} - \frac{g}{k(g + kv)}\,\mathrm{d}v\) Split for integration | M1 A1 |
| \(H = \left[\dfrac{-v}{k} + \dfrac{g}{k^2}\ln(g + kv)\right]_U^0\) | A1 |
| \(= \dfrac{U}{k} - \dfrac{g}{k^2}\ln\left(\dfrac{g + kU}{g}\right)\) Use of limits | M1 A1 |
| (8 marks) |