2 In trials for a vehicle emergency stopping system, a small car of mass 400 kg is propelled towards a buffer. The buffer is modelled as a light spring of stiffness \(5000 \mathrm {~N} \mathrm {~m} ^ { - 1 }\). One end of the spring is fixed, and the other end points directly towards the oncoming car. Throughout this question, there is no driving force acting on the car, and there are no resistances to motion apart from those specifically mentioned.
At first, the buffer is mounted on a horizontal surface, and the car has speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it hits the buffer, as shown in Fig. 2.1.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3ec81c4e-e0fa-43d9-9c79-ef9df746be8f-3_220_1105_671_520}
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\caption{Fig. 2.1}
\end{figure}
- Find the compression of the spring when the car comes (instantaneously) to rest.
The buffer is now mounted on a slope making an angle \(\theta\) with the horizontal, where \(\sin \theta = \frac { 1 } { 7 }\). The car is released from rest and travels 7.35 m down the slope before hitting the buffer, as shown in Fig. 2.2.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3ec81c4e-e0fa-43d9-9c79-ef9df746be8f-3_268_1091_1329_529}
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\caption{Fig. 2.2}
\end{figure} - Verify that the car comes instantaneously to rest when the spring is compressed by 1.4 m .
The surface of the slope (including the section under the buffer) is now covered with gravel which exerts a constant resistive force of 7560 N on the car. The car is moving down the slope, and has speed \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is 24 m from the buffer, as shown in Fig. 2.3. It comes to rest when the spring has been compressed by \(x\) metres.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3ec81c4e-e0fa-43d9-9c79-ef9df746be8f-3_305_1087_2122_529}
\captionsetup{labelformat=empty}
\caption{Fig. 2.3}
\end{figure} - By considering work and energy, find the value of \(x\).