6 A particle moves in a straight line. It starts from rest at a fixed point \(O\) on the line. Its acceleration at time \(t \mathrm {~s}\) after leaving \(O\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = 0.4 t ^ { 3 } - 4.8 t ^ { \frac { 1 } { 2 } }\).

1. Show that, in the subsequent motion, the acceleration of the particle when it comes to instantaneous rest is \(16 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the displacement of the particle from \(O\) at \(t = 5\).
   \includegraphics[max width=\textwidth, alt={}, center]{06df8c0d-dd38-4e3b-b1a4-72120a81050e-12_554_878_260_635} The diagram shows the vertical cross-section \(P Q R\) of a slide. The part \(P Q\) is a straight line of length 8 m inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.8\). The straight part \(P Q\) is tangential to the curved part \(Q R\), and \(R\) is \(h \mathrm {~m}\) above the level of \(P\). The straight part \(P Q\) of the slide is rough and the curved part \(Q R\) is smooth. A particle of mass 0.25 kg is projected with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from \(P\) towards \(Q\) and comes to rest at \(R\). The coefficient of friction between the particle and \(P Q\) is 0.5 .
3. Find the work done by the friction force during the motion of the particle from \(P\) to \(Q\).
4. Hence find the speed of the particle at \(Q\).
5. Find the value of \(h\).
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