Body collecting atmospheric moisture

7. At time \(t = 0\), a small body is projected vertically upwards. While ascending it picks up small drops of moisture from the atmosphere. The drops of moisture are at rest before they are picked up. At time \(t\), the combined body \(P\) has mass \(m\) and speed \(v\).

1. Show that, while \(P\) is moving upwards, \(m \frac { \mathrm {~d} v } { \mathrm {~d} t } + v \frac { \mathrm {~d} m } { \mathrm {~d} t } = - m g\). The initial mass of \(P\) is \(M\), and \(m = M \mathrm { e } ^ { k t }\), where \(k\) is a positive constant.
2. Show that, while \(P\) is moving upwards, \(\frac { \mathrm { d } } { \mathrm { d } t } \left( v \mathrm { e } ^ { k t } \right) = - g \mathrm { e } ^ { k t }\). Given that the initial projection speed of \(P\) is \(\frac { g } { 2 k }\),
3. find, in terms of \(M\), the mass of \(P\) when it reaches its highest point.
   (Total 15 marks)

A raindrop falls vertically under gravity through a cloud. In a model of the motion the raindrop is assumed to be spherical at all times and the cloud is assumed to consist of stationary water particles. At time \(t = 0\), the raindrop is at rest and has radius \(a\). As the raindrop falls, water particles from the cloud condense onto it and the radius of the raindrop is assumed to increase at a constant rate \(\lambda\). At time \(t\) the speed of the raindrop is \(v\).

1. Show that $$\frac{dv}{dt} + \frac{3\lambda v}{(\lambda t + a)} = g.$$ [8]

1. Find the speed of the raindrop when its radius is \(3a\). [7]

As a hailstone falls under gravity in still air, its mass increases. At time \(t\) the mass of the hailstone is \(m\). The hailstone is modelled as a uniform sphere of radius \(r\) such that $$\frac{dr}{dt} = kr,$$ where \(k\) is a positive constant.

1. Show that \(\frac{dm}{dt} = 3km\). [2]

Assuming that there is no air resistance,

1. show that the speed \(v\) of the hailstone at time \(t\) satisfies $$\frac{dv}{dt} = g - 3kv.$$ [4]

Given that the speed of the hailstone at time \(t = 0\) is \(u\),

1. find an expression for \(v\) in terms of \(t\). [5]
2. Hence show that the speed of the hailstone approaches the limiting value \(\frac{g}{3k}\). [1]