Edexcel M1 (Mechanics 1) 2012 January

1. A railway truck \(P\), of mass \(m \mathrm {~kg}\), is moving along a straight horizontal track with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Truck \(P\) collides with a truck \(Q\) of mass 3000 kg , which is at rest on the same track. Immediately after the collision the speed of \(P\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(Q\) is \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The direction of motion of \(P\) is reversed by the collision.

Modelling the trucks as particles, find

1. the magnitude of the impulse exerted by \(P\) on \(Q\),
2. the value of \(m\).

2. A car of mass 1000 kg is towing a caravan of mass 750 kg along a straight horizontal road. The caravan is connected to the car by a tow-bar which is parallel to the direction of motion of the car and the caravan. The tow-bar is modelled as a light rod. The engine of the car provides a constant driving force of 3200 N . The resistances to the motion of the car and the caravan are modelled as constant forces of magnitude 800 newtons and \(R\) newtons respectively. Given that the acceleration of the car and the caravan is \(0.88 \mathrm {~ms} ^ { - 2 }\),

1. show that \(R = 860\),
2. find the tension in the tow-bar.

3. Three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) acting on a particle \(P\) are given by $$\begin{aligned} & \mathbf { F } _ { 1 } = ( 7 \mathbf { i } - 9 \mathbf { j } ) \mathrm { N }
& \mathbf { F } _ { 2 } = ( 5 \mathbf { i } + 6 \mathbf { j } ) \mathrm { N }
& \mathbf { F } _ { 3 } = ( p \mathbf { i } + q \mathbf { j } ) \mathrm { N } \end{aligned}$$ where \(p\) and \(q\) are constants.
Given that \(P\) is in equilibrium,

1. find the value of \(p\) and the value of \(q\). The force \(\mathbf { F } _ { 3 }\) is now removed. The resultant of \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) is \(\mathbf { R }\). Find
2. the magnitude of \(\mathbf { R }\),
3. the angle, to the nearest degree, that the direction of \(\mathbf { R }\) makes with \(\mathbf { j }\).

4. \begin{figure}[h] \end{figure} A non-uniform \(\operatorname { rod } A B\), of mass \(m\) and length \(5 d\), rests horizontally in equilibrium on two supports at \(C\) and \(D\), where \(A C = D B = d\), as shown in Figure 1. The centre of mass of the rod is at the point \(G\). A particle of mass \(\frac { 5 } { 2 } m\) is placed on the rod at \(B\) and the rod is on the point of tipping about \(D\).

1. Show that \(G D = \frac { 5 } { 2 } d\). The particle is moved from \(B\) to the mid-point of the rod and the rod remains in equilibrium.
2. Find the magnitude of the normal reaction between the support at \(D\) and the rod.

1. A stone is projected vertically upwards from a point \(A\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After projection the stone moves freely under gravity until it returns to \(A\). The time between the instant that the stone is projected and the instant that it returns to \(A\) is \(3 \frac { 4 } { 7 }\) seconds.

Modelling the stone as a particle,

1. show that \(u = 17 \frac { 1 } { 2 }\),
2. find the greatest height above \(A\) reached by the stone,
3. find the length of time for which the stone is at least \(6 \frac { 3 } { 5 } \mathrm {~m}\) above \(A\).

1. A car moves along a straight horizontal road from a point \(A\) to a point \(B\), where \(A B = 885 \mathrm {~m}\). The car accelerates from rest at \(A\) to a speed of \(15 \mathrm {~ms} ^ { - 1 }\) at a constant rate \(a \mathrm {~ms} ^ { - 2 }\). The time for which the car accelerates is \(\frac { 1 } { 3 } T\) seconds. The car maintains the speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for \(T\) seconds. The car then decelerates at a constant rate of \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) stopping at \(B\).
   1. Find the time for which the car decelerates.
   2. Sketch a speed-time graph for the motion of the car.
   3. Find the value of \(T\).
   4. Find the value of \(a\).
   5. Sketch an acceleration-time graph for the motion of the car.
      

7. [In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are due east and due north respectively. Position vectors are relative to a fixed origin \(O\).] A boat \(P\) is moving with constant velocity \(( - 4 \mathbf { i } + 8 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\).

1. Calculate the speed of \(P\). When \(t = 0\), the boat \(P\) has position vector \(( 2 \mathbf { i } - 8 \mathbf { j } ) \mathrm { km }\). At time \(t\) hours, the position vector of \(P\) is \(\mathbf { p ~ k m }\).
2. Write down \(\mathbf { p }\) in terms of \(t\). A second boat \(Q\) is also moving with constant velocity. At time \(t\) hours, the position vector of \(Q\) is \(\mathbf { q } \mathrm { km }\), where $$\mathbf { q } = 18 \mathbf { i } + 12 \mathbf { j } - t ( 6 \mathbf { i } + 8 \mathbf { j } )$$ Find
3. the value of \(t\) when \(P\) is due west of \(Q\),
4. the distance between \(P\) and \(Q\) when \(P\) is due west of \(Q\).

8. \begin{figure}[h] \end{figure} A particle \(P\) of mass 4 kg is moving up a fixed rough plane at a constant speed of \(16 \mathrm {~ms} ^ { - 1 }\) under the action of a force of magnitude 36 N . The plane is inclined at \(30 ^ { \circ }\) to the horizontal. The force acts in the vertical plane containing the line of greatest slope of the plane through \(P\), and acts at \(30 ^ { \circ }\) to the inclined plane, as shown in Figure 2. The coefficient of friction between \(P\) and the plane is \(\mu\). Find

1. the magnitude of the normal reaction between \(P\) and the plane,
2. the value of \(\mu\). The force of magnitude 36 N is removed.
3. Find the distance that \(P\) travels between the instant when the force is removed and the instant when it comes to rest.