7 A box B of mass \(m \mathrm {~kg}\) is raised vertically by an engine working at a constant rate of \(k m g \mathrm {~W}\). Initially B is at rest. The speed of B when it has been raised a distance \(x \mathrm {~m}\) is denoted by \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
- Show that \(v ^ { 2 } \frac { d v } { d x } = ( k - v ) g\).
- Verify that \(\mathrm { gx } = \mathrm { k } ^ { 2 } \ln \left( \frac { \mathrm { k } } { \mathrm { k } - \mathrm { v } } \right) - \mathrm { kv } - \frac { 1 } { 2 } \mathrm { v } ^ { 2 }\).
- By using the work-energy principle, show that the time taken for B to reach a speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from rest is given by
\(\frac { \mathrm { k } } { \mathrm { g } } \ln \left( \frac { \mathrm { k } } { \mathrm { k } - \mathrm { V } } \right) - \frac { \mathrm { V } } { \mathrm { g } }\).