- A particle \(P\) of mass 0.5 kg is moving along the positive \(x\)-axis in the direction of \(x\) increasing. At time \(t\) seconds \(( t \geqslant 0 ) , P\) is \(x\) metres from the origin \(O\) and the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resultant force acting on \(P\) is directed towards \(O\) and has magnitude \(k v ^ { 2 } \mathrm {~N}\), where \(k\) is a positive constant.
When \(x = 1 , v = 4\) and when \(x = 2 , v = 2\)
- Show that \(v = a b ^ { x }\), where \(a\) and \(b\) are constants to be found.
The time taken for the speed of \(P\) to decrease from \(4 \mathrm {~ms} ^ { - 1 }\) to \(2 \mathrm {~ms} ^ { - 1 }\) is \(T\) seconds.
- Show that \(T = \frac { 1 } { 4 \ln 2 }\)