- A particle, \(P\), of mass 0.4 kg is moving along the positive \(x\)-axis, in the positive \(x\) direction under the action of a single force. At time \(t\) seconds, \(t > 0 , P\) is \(x\) metres from the origin \(O\) and the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The force is acting in the direction of \(x\) increasing and has magnitude \(\frac { k } { v }\) newtons, where \(k\) is a constant.
At \(x = 3 , v = 2\) and at \(x = 6 , v = 2.5\)
- Show that \(v ^ { 3 } = \frac { 61 x + 9 } { 24 }\)
The time taken for the speed of \(P\) to increase from \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is \(T\) seconds.
- Use algebraic integration to show that \(T = \frac { 81 } { 61 }\)