3.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bab18666-3571-4906-ab2f-b85e87c6e8f4-4_289_846_358_678}
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\caption{Figure 1}
\end{figure}
Figure 1 represents the plan of part of a smooth horizontal floor, where \(W _ { 1 }\) and \(W _ { 2 }\) are two fixed parallel vertical walls. The walls are 3 metres apart.
A particle lies at rest at a point \(O\) on the floor between the two walls, where the point \(O\) is \(d\) metres, \(0 < d \leqslant 3\), from \(W _ { 1 }\)
At time \(t = 0\), the particle is projected from \(O\) towards \(W _ { 1 }\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction perpendicular to the walls.
The coefficient of restitution between the particle and each wall is \(\frac { 2 } { 3 }\)
The particle returns to \(O\) at time \(t = T\) seconds, having bounced off each wall once.
- Show that \(T = \frac { 45 - 5 d } { 4 u }\)
The value of \(u\) is fixed, the particle still hits each wall once but the value of \(d\) can now vary.
- Find the least possible value of \(T\), giving your answer in terms of \(u\). You must give a reason for your answer.
\section*{4.}
A car of mass 600 kg pulls a trailer of mass 150 kg along a straight horizontal road. The trailer is connected to the car by a light inextensible towbar, which is parallel to the direction of motion of the car. The resistance to the motion of the trailer is modelled as a constant force of magnitude 200 N . At the instant when the speed of the car is \(v \mathrm {~ms} ^ { - 1 }\), the resistance to the motion of the car is modelled as a force of magnitude \(( 200 + \lambda v ) \mathrm { N }\), where \(\lambda\) is a constant.
When the engine of the car is working at a constant rate of 15 kW , the car is moving at a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
- Show that \(\lambda = 8\)
Later on, the car is pulling the trailer up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 15 }\)
The resistance to the motion of the trailer from non-gravitational forces is modelled as a constant force of magnitude 200 N at all times. At the instant when the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the car from non-gravitational forces is modelled as a force of magnitude \(( 200 + 8 v ) \mathrm { N }\).
The engine of the car is again working at a constant rate of 15 kW .
When \(v = 10\), the towbar breaks. The trailer comes to instantaneous rest after moving a distance \(d\) metres up the road from the point where the towbar broke. - Find the acceleration of the car immediately after the towbar breaks.
- Use the work-energy principle to find the value of \(d\).