3.03q Dynamics: motion under force in plane

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CAIE M1 2020 Specimen Q5
9 marks Standard +0.3
5 A car of mass 1200 kg is pulling a trailer of mass 800 kg up a hill inclined at an angle of \(\sin ^ { - 1 } ( 0.1 )\) to the horizontal. The car and the trailer are connected by a light rigid tow-bar which is parallel to the road. The driving force of the car's engine is 2500 N and the resistances to the car and trailer are 300 N and 100 N respectively.
  1. Find the acceleration of the system and the tension in the tow-bar.
  2. When the car and trailer are travelling at a speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the driving force becomes zero. Find the time, in seconds, before the system comes to rest and the force in the tow-bar during this time.
CAIE M1 2016 March Q5
7 marks Standard +0.3
5 A car of mass 1200 kg is pulling a trailer of mass 800 kg up a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.1\). The system of the car and the trailer is modelled as two particles connected by a light inextensible cable. The driving force of the car's engine is 2500 N and the resistances to the car and trailer are 100 N and 150 N respectively.
  1. Find the acceleration of the system and the tension in the cable.
  2. When the car and trailer are travelling at a speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the driving force becomes zero. The cable remains taut. Find the time, in seconds, before the system comes to rest.
Edexcel M1 2009 January Q7
16 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{86bb11a4-b409-49b1-bffb-d0e3727d345c-11_495_892_301_523} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} One end of a light inextensible string is attached to a block \(P\) of mass 5 kg . The block \(P\) is held at rest on a smooth fixed plane which is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 3 } { 5 }\). The string lies along a line of greatest slope of the plane and passes over a smooth light pulley which is fixed at the top of the plane. The other end of the string is attached to a light scale pan which carries two blocks \(Q\) and \(R\), with block \(Q\) on top of block \(R\), as shown in Figure 3. The mass of block \(Q\) is 5 kg and the mass of block \(R\) is 10 kg . The scale pan hangs at rest and the system is released from rest. By modelling the blocks as particles, ignoring air resistance and assuming the motion is uninterrupted, find
    1. the acceleration of the scale pan,
    2. the tension in the string,
  2. the magnitude of the force exerted on block \(Q\) by block \(R\),
  3. the magnitude of the force exerted on the pulley by the string.
CAIE M1 2024 June Q5
11 marks Standard +0.3
5 A van of mass 4500 kg is towing a trailer of mass 750 kg down a straight hill inclined at an angle of \(\theta\) to the horizontal where \(\sin \theta = 0.05\). The van and the trailer are connected by a light rigid tow-bar which is parallel to the road. There are constant resistance forces of 2500 N on the van and 300 N on the trailer.
  1. It is given that the tension in the tow-bar is 450 N . Find the acceleration of the trailer and the driving force of the van's engine.
    On another occasion, the van and trailer ascend a straight hill inclined at an angle of \(\alpha\) to the horizontal where \(\sin \alpha = 0.09\). The driving force of the van's engine is now 9100 N , and the speed of the van at the bottom of the hill is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistances to motion are unchanged.
    1. Find the acceleration of the van and the tension in the tow-bar.
    2. Find the speed of the van when it has travelled a distance of 375 m up the hill.