Edexcel M1 (Mechanics 1) 2018 June

1. Particle \(P\) has mass \(3 m\) and particle \(Q\) has mass \(m\). The particles are moving towards each other in opposite directions along the same straight line on a smooth horizontal plane. The particles collide directly. Immediately before the collision the speed of \(P\) is \(u\) and the speed of \(Q\) is \(3 u\). In the collision, the magnitude of the impulse exerted by \(Q\) on \(P\) is \(5 m u\).
   1. Find the speed of \(P\) immediately after the collision.
   2. Find the speed of \(Q\) immediately after the collision.
      

2. \begin{figure}[h] \end{figure} A uniform wooden beam \(A B\), of mass 20 kg and length 4 m , rests in equilibrium in a horizontal position on two supports. One support is at \(C\), where \(A C = 1.6 \mathrm {~m}\), and the other support is at \(D\), where \(D B = 0.4 \mathrm {~m}\). A boy of mass 60 kg stands on the beam at the point \(P\), where \(A P = 3 \mathrm {~m}\), as shown in Figure 1. The beam remains in equilibrium in a horizontal position. By modelling the boy as a particle and the beam as a uniform rod,

1. 1. find, in terms of \(g\), the magnitude of the force exerted on the beam by the support at \(C\),
   2. find, in terms of \(g\), the magnitude of the force exerted on the beam by the support at \(D\). The boy now starts to walk slowly along the beam towards the end \(A\).
2. Find the greatest distance he can walk from \(P\) without the beam tilting.

3. A cyclist starts from rest at the point \(O\) on a straight horizontal road. The cyclist moves along the road with constant acceleration \(2 \mathrm {~ms} ^ { - 2 }\) for 4 seconds and then continues to move along the road at constant speed. At the instant when the cyclist stops accelerating, a motorcyclist starts from rest at the point \(O\) and moves along the road with constant acceleration \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) in the same direction as the cyclist. The motorcyclist has been moving for \(T\) seconds when she overtakes the cyclist.

1. Sketch, on the same axes, a speed-time graph for the motion of the cyclist and a speed-time graph for the motion of the motorcyclist, to the time when the motorcyclist overtakes the cyclist.
2. Find, giving your answer to 1 decimal place, the value of \(T\).

4. A rough plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). A particle of mass 2 kg is projected with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) on the plane, up a line of greatest slope of the plane. The coefficient of friction between the particle and the plane is 0.25

1. Find the magnitude of the frictional force acting on the particle as it moves up the plane. The particle comes to instantaneous rest at the point \(A\).
2. Find the distance \(O A\). The particle now moves down the plane from \(A\).
3. Find the speed of \(P\) as it passes through \(O\).

5. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular horizontal unit vectors and position vectors are given relative to a fixed origin \(O\).] A particle \(P\) is moving in a straight line with constant velocity. At 9 am, the position vector of \(P\) is \(( 7 \mathbf { i } + 5 \mathbf { j } ) \mathrm { km }\) and at 9.20 am , the position vector of \(P\) is \(6 \mathbf { i } \mathrm {~km}\). At time \(t\) hours after 9 am , the position vector of \(P\) is \(\mathbf { r } _ { P } \mathrm {~km}\).

1. Find, in \(\mathrm { kmh } ^ { - 1 }\), the speed of \(P\).
2. Show that \(\mathbf { r } _ { P } = ( 7 - 3 t ) \mathbf { i } + ( 5 - 15 t ) \mathbf { j }\).
3. Find the value of \(t\) when \(\mathbf { r } _ { P }\) is parallel to \(16 \mathbf { i } + 5 \mathbf { j }\). The position vector of another particle \(Q\), at time \(t\) hours after 9 am , is \(\mathbf { r } _ { Q } \mathrm {~km}\), where \(\mathbf { r } _ { Q } = ( 5 + 2 t ) \mathbf { i } + ( - 3 + 5 t ) \mathbf { j }\)
4. Show that \(P\) and \(Q\) will collide and find the position vector of the point of collision.

6. A car pulls a trailer along a straight horizontal road using a light inextensible towbar. The mass of the car is \(M \mathrm {~kg}\), the mass of the trailer is 600 kg and the towbar is horizontal and parallel to the direction of motion. There is a resistance to motion of magnitude 200 N acting on the car and a resistance to motion of magnitude 100 N acting on the trailer. The driver of the car spots a hazard ahead. Instantly he reduces the force produced by the engine of the car to zero and applies the brakes of the car. The brakes produce a braking force on the car of magnitude 6500 N and the car and the trailer have a constant deceleration of magnitude \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) Given that the resistances to motion on the car and trailer are unchanged and that the car comes to rest after travelling 40.5 m from the point where the brakes were applied, find

1. the thrust in the towbar while the car is braking,
2. the value of \(M\),
3. the time it takes for the car to stop after the brakes are applied.

7. \begin{figure}[h] \end{figure} A washing line \(A B C D\) is fixed at the points \(A\) and \(D\). There are two heavy items of clothing hanging on the washing line, one fixed at \(B\) and the other fixed at \(C\). The washing line is modelled as a light inextensible string, the item at \(B\) is modelled as a particle of mass 3 kg and the item at \(C\) is modelled as a particle of mass \(M \mathrm {~kg}\). The section \(A B\) makes an angle \(\alpha\) with the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\), the section \(B C\) is horizontal and the section \(C D\) makes an angle \(\beta\) with the horizontal, where \(\tan \beta = \frac { 12 } { 5 }\), as shown in Figure 2. The system is in equilibrium.

1. Find the tension in \(A B\).
2. Find the tension in BC.
3. Find the value of \(M\).
   

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