6.02a Work done: concept and definition

A particle \(P\) of mass \(m\) moves along the positive \(x\)-axis. When its displacement from the origin \(O\) is \(x\) its velocity is \(v\), where \(v \geqslant 0\). It is subject to two forces: a constant force \(T\) in the positive \(x\) direction, and a resistive force which is proportional to \(v^2\).

1. Show that \(v^2 = \frac{1}{k}\left(T - Ae^{-\frac{2kx}{m}}\right)\) where \(A\) and \(k\) are constants. [5]

\(P\) starts from rest at \(O\).

1. Find an expression for the work done against the resistance to motion as \(P\) moves from \(O\) to the point where \(x = 1\). [4]
2. Find an expression for the limiting value of the velocity of \(P\) as \(x\) increases. [1]

A stone that weighs 15 kg is propelled across the ice in an ice rink with an initial speed of \(4 \text{ m s}^{-1}\). The coefficient of friction between the stone and the ice is \(0.017\). How far does the stone slide before it comes to rest? [5]

A particle of mass \(0.01\) kg is projected vertically upwards from a point \(G\) at ground level with speed \(165 \text{ m s}^{-1}\) and reaches a maximum height of \(1237.5\) m. Throughout its motion it experiences a constant resistance.

1. Find the acceleration of the particle as it ascends and hence the magnitude of the resistance. [4]
2. During its descent back to \(G\) the particle experiences the same constant resistance. Find the time taken for the descent. [4]