3. A particle \(P\) of mass 0.5 kg moves in a straight line with simple harmonic motion, completing 4 oscillations per second. The particle comes to instantaneous rest at the fixed points \(A\) and \(B\), where \(A B = 0.5 \mathrm {~m}\).
- Find the maximum magnitude of the acceleration of \(P\).
When \(P\) is moving at its maximum speed it receives an impulse. The direction of this impulse is opposite to the direction in which \(P\) is moving when it receives the impulse. The impulse causes \(P\) to reverse its direction of motion but \(P\) continues to move with simple harmonic motion. The centre and period of this new simple harmonic motion are the same as the centre and period of the original simple harmonic motion. The amplitude is now half the original amplitude.
- Find the magnitude of the impulse.
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