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A particle \(P\) of mass 0.5 kg is projected along a smooth horizontal surface towards a fixed point \(A\). Initially \(P\) is at a point \(O\) on the surface, and after projection, \(P\) has a displacement from \(O\) of \(x \mathrm {~m}\) and velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The particle \(P\) is connected to \(A\) by a light elastic string of natural length 0.8 m and modulus of elasticity 16 N . The distance \(O A\) is 1.6 m (see diagram). The motion of \(P\) is resisted by a force of magnitude \(24 x ^ { 2 } \mathrm {~N}\).
- Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 32 - 40 x - 48 x ^ { 2 }\) while \(P\) is in motion and the string is stretched.
The maximum value of \(v\) is 4.5 . - Find the initial value of \(v\).