- A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds, \(t \geqslant 0 , P\) is \(x\) metres from the origin \(O\) and moving with velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of \(x\) increasing, where
$$v = 5 \sin 2 t$$
When \(t = 0 , x = 1\) and \(P\) is at rest.
- Find the magnitude and direction of the acceleration of \(P\) at the instant when \(P\) is next at rest.
- Show that \(1 \leqslant x \leqslant 6\)
- Find the total time, in the first \(4 \pi\) seconds of the motion, for which \(P\) is more than 3 metres from \(O\)
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