6 A particle moves along a straight line through an origin O . Initially the particle is at O .
At time \(t \mathrm {~s}\), its displacement from O is \(x \mathrm {~m}\) and its velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by
$$v = 24 - 18 t + 3 t ^ { 2 } .$$
- Find the times, \(T _ { 1 } \mathrm {~s}\) and \(T _ { 2 } \mathrm {~s}\) (where \(T _ { 2 } > T _ { 1 }\) ), at which the particle is stationary.
- Find an expression for \(x\) at time \(t \mathrm {~s}\).
Show that when \(t = T _ { 1 } , x = 20\) and find the value of \(x\) when \(t = T _ { 2 }\).
Section B (36 marks)
\(7 \quad\) Abi and Bob are standing on the ground and are trying to raise a small object of weight 250 N to the top of a building. They are using long light ropes. Fig. 7.1 shows the initial situation.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{83e69140-4abf-4713-85da-922ce7530e47-4_773_1071_429_497}
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\caption{Fig. 7.1}
\end{figure}
Abi pulls vertically downwards on the rope A with a force \(F\) N. This rope passes over a small smooth pulley and is then connected to the object. Bob pulls on another rope, B, in order to keep the object away from the side of the building.
In this situation, the object is stationary and in equilibrium. The tension in rope B, which is horizontal, is 25 N . The pulley is 30 m above the object. The part of rope A between the pulley and the object makes an angle \(\theta\) with the vertical.