Edexcel M1 (Mechanics 1) 2020 January

1. Two particles, \(P\) and \(Q\), of mass \(m _ { 1 }\) and \(m _ { 2 }\) respectively, are moving on a smooth horizontal plane. The particles are moving towards each other in opposite directions along the same straight line when they collide directly. Immediately before the collision, both particles are moving with speed \(u\).

The direction of motion of each particle is reversed by the collision.
Immediately after the collision, the speed of \(Q\) is \(\frac { 1 } { 3 } u\).

1. Find, in terms of \(m _ { 2 }\) and \(u\), the magnitude of the impulse exerted by \(P\) on \(Q\) in the collision.
2. Find, in terms of \(m _ { 1 } , m _ { 2 }\) and \(u\), the speed of \(P\) immediately after the collision.
3. Hence show that \(m _ { 2 } > \frac { 3 } { 4 } m _ { 1 }\)

2. \begin{figure}[h] \end{figure} A non-uniform beam \(A B\) has length 6 m and weight \(W\) newtons. The beam is supported in equilibrium in a horizontal position by two vertical ropes, one attached to the beam at \(A\) and the other attached to the beam at \(C\), where \(C B = 1.5 \mathrm {~m}\), as shown in Figure 1 . The centre of mass of the beam is 2.625 m from \(A\). The ropes are modelled as light strings. The beam is modelled as a non-uniform rod. Given that the tension in the rope attached at \(C\) is 20 N greater than the tension in the rope attached at \(A\),

1. find the value of \(W\).
2. State how you have used the fact that the beam is modelled as a rod.
   

3. A particle, \(P\), is projected vertically upwards with speed \(U\) from a fixed point \(O\). At the instant when \(P\) reaches its greatest height \(H\) above \(O\), a second particle, \(Q\), is projected with speed \(\frac { 1 } { 2 } U\) vertically upwards from \(O\).

1. Find \(H\) in terms of \(U\) and \(g\).
2. Find, in terms of \(U\) and \(g\), the time between the instant when \(Q\) is projected and the instant when the two particles collide.
3. Find where the two particles collide. DO NOT WRITEIN THIS AREA
   \includegraphics[max width=\textwidth, alt={}, center]{916543cb-14f7-486c-ba3c-eda9be134045-08_2666_99_107_1957}

4. \begin{figure}[h] \end{figure} Two identical small rings, \(A\) and \(B\), each of mass \(m\), are threaded onto a rough horizontal wire. The rings are connected by a light inextensible string. A particle \(C\) of mass \(3 m\) is attached to the midpoint of the string. The particle \(C\) hangs in equilibrium below the wire with angle \(B A C = \beta\), as shown in Figure 2. The tension in each of the parts, \(A C\) and \(B C\), of the string is \(T\)

1. By considering particle \(C\), find \(T\) in terms of \(m , g\) and \(\beta\)
2. Find, in terms of \(m\) and \(g\), the magnitude of the normal reaction between the wire and \(A\). The coefficient of friction between each ring and the wire is \(\frac { 4 } { 5 }\)
   The two rings, \(A\) and \(B\), are on the point of sliding along the wire towards each other.
3. Find the value of \(\tan \beta\)
   \includegraphics[max width=\textwidth, alt={}, center]{916543cb-14f7-486c-ba3c-eda9be134045-11_2255_50_314_34}
   

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5. A car travels at a constant speed of \(40 \mathrm {~ms} ^ { - 1 }\) in a straight line along a horizontal racetrack. At time \(t = 0\), the car passes a motorcyclist who is at rest. The motorcyclist immediately sets off to catch up with the car. The motorcyclist accelerates at \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 15 s and then accelerates at \(1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for a further \(T\) seconds until he catches up with the car.

1. Sketch, on the same axes, the speed-time graph for the motion of the car and the speed-time graph for the motion of the motorcyclist, from time \(t = 0\) to the instant when the motorcyclist catches up with the car. At the instant when \(t = t _ { 1 }\) seconds, the car and the motorcyclist are moving at the same speed.
2. Find the value of \(t _ { 1 }\)
3. Show that \(T ^ { 2 } + k T - 300 = 0\), where \(k\) is a constant to be found. DO NOT WRITEIN THIS AREA
   

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6. A force \(\mathbf { F }\) is given by \(\mathbf { F } = ( 10 \mathbf { i } + \mathbf { j } ) \mathrm { N }\).

1. Find the exact value of the magnitude of \(\mathbf { F }\).
2. Find, in degrees, the size of the angle between the direction of \(\mathbf { F }\) and the direction of the vector \(( \mathbf { i } + \mathbf { j } )\). The resultant of the force \(\mathbf { F }\) and the force \(( - 15 \mathbf { i } + a \mathbf { j } ) \mathrm { N }\), where \(a\) is a constant, is parallel to, but in the opposite direction to, the vector \(( 2 \mathbf { i } - 3 \mathbf { j } )\).
3. Find the value of \(a\). \includegraphics[max width=\textwidth, alt={}, center]{916543cb-14f7-486c-ba3c-eda9be134045-19_104_59_2613_1886}

7. \begin{figure}[h] \end{figure} A particle \(A\) of mass 4 kg is held at rest on a rough horizontal table. Particle \(A\) is attached to one end of a string that passes over a pulley \(P\). The pulley is fixed the the of the table. The other end of the string is attached to a particle \(B\), of mass 3 kg , which hangs freely below \(P\). The part of the string from \(A\) to \(P\) is perpendicular to the edge of the table and \(A , P\) and \(B\) all lie in the same vertical plane. The string is modelled as being light and inextensible and the pulley is modelled as being small, smooth and light. The system is released from rest with the string taut. At the instant of release, \(A\) is 2 m from the edge of the table and \(B\) is 1.4 m above a horizontal floor, as shown in Figure 3. After descending with constant acceleration for 2 seconds, \(B\) hits the floor and does not rebound.

1. Show that the acceleration of \(A\) before \(B\) hits the floor is \(0.7 \mathrm {~ms} ^ { - 2 }\)
2. State which of the modelling assumptions you have used in order to answer part (a).
3. Find the magnitude of the resultant force exerted on the pulley by the string. The coefficient of friction between \(A\) and the table is \(\mu\).
4. Find the value of \(\mu\).
5. Determine, by calculation, whether or not \(A\) reaches the pulley. DO NOT WRITEIN THIS AREA
   

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   \includegraphics[max width=\textwidth, alt={}, center]{916543cb-14f7-486c-ba3c-eda9be134045-23_2255_50_314_34}