Show that \(B\) 's motion can be modelled by the differential equation \(\frac { 1 } { \mathrm { v } } \frac { \mathrm { dv } } { \mathrm { dx } } = - 4\).
Solve the differential equation in part (a) to find the particular solution for \(v\) in terms of \(x\) and \(u\).
By considering the behaviour of \(v\) as \(x \longrightarrow \infty\) describe one feature of the model that is not realistic.
At the instant when \(B\) reaches the point \(A\), where \(\mathrm { x } = \mathrm { X }\), its speed is \(V \mathrm {~ms} ^ { - 1 }\). The work done by the resistance as \(B\) moves from \(O\) to \(A\) is denoted by \(W \mathrm {~J}\).
Use the formula \(\mathrm { W } = \int \mathrm { F } \mathrm { dx }\) to determine an expression for \(W\) in terms of \(X\) and \(u\).
Explain the relevance of the sign of your answer in part (c)(i).
By writing your answer to part (c)(i) in terms of \(V\) and \(u\) show how the quantity \(W\) relates to the energy of \(B\).