Edexcel M1 (Mechanics 1) 2021 January

1. A small stone is projected vertically upwards with speed \(20 \mathrm {~ms} ^ { - 1 }\) from a point \(O\) which is 5 m above horizontal ground. The stone is modelled as a particle moving freely under gravity.

Find

1. the speed of the stone at the instant when it is 2 m above the ground,
2. the total time between the instant when the stone is projected from \(O\) and the instant when it first strikes the ground.

2. Two particles, \(P\) and \(Q\), have masses \(2 m\) and \(m\) respectively. 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 \(3 u\) and the speed of \(Q\) is \(2 u\). The magnitude of the impulse exerted on \(Q\) by \(P\) in the collision is 5mu. Find

1. the speed of \(P\) immediately after the collision,
2. the speed of \(Q\) immediately after the collision.

3. \begin{figure}[h] \end{figure} A parcel of mass 20 kg is at rest on a rough horizontal floor. The coefficient of friction between the parcel and the floor is 0.3 Two forces, both acting in the same vertical plane, of magnitudes 200 N and \(T \mathrm {~N}\) are applied to the parcel. The line of action of the 200 N force makes an angle of \(15 ^ { \circ }\) with the horizontal and the line of action of the \(T \mathrm {~N}\) force makes an angle of \(25 ^ { \circ }\) with the horizontal, as shown in Figure 1. The parcel is modelled as a particle \(P\). Find the smallest value of \(T\) for which \(P\) remains in equilibrium.

4. \begin{figure}[h] \end{figure} \begin{verbatim} A metal girder \(A B\) has weight \(W\) newtons and length 6 m . The girder rests in a horizontal position on two supports \(C\) and \(D\) where \(A C = D B = 1 \mathrm {~m}\), as shown in Figure 2. When a force of magnitude 900 N is applied vertically upwards to the girder at \(A\), the girder is about to tilt about \(D\). When a force of magnitude 1500 N is applied vertically upwards to the girder at \(B\), the girder is about to tilt about \(C\). The girder is modelled as a non-uniform rod whose centre of mass is a distance \(x\) metres from \(A\). Find the value of \(x\). A metal girder AB has weight When a force of magnitude 1500 N is applied vertically upwards to the girder at \(B\), the girder is about to tilt about \(C\). The girder is modelled as a non-uniform rod whose centre of mass is a distance \(x\) metres from \(A\). Find the value of \(x\). \end{verbatim}

5. A particle is acted upon by two forces \(\mathbf { F }\) and \(\mathbf { G }\). The force \(\mathbf { F }\) has magnitude 8 N and acts in a direction with a bearing of \(240 ^ { \circ }\). The force \(\mathbf { G }\) has magnitude 10 N and acts due South. Given that \(\mathbf { R } = \mathbf { F } + \mathbf { G }\), find

1. the magnitude of \(\mathbf { R }\),
2. the direction of \(\mathbf { R }\), giving your answer as a bearing to the nearest degree. in a direction with a bearing of \(240 ^ { \circ }\). The force \(\mathbf { G }\) has magnitude 10 N and acts due South. Given that \(\mathbf { R } = \mathbf { F } + \mathbf { G }\), find

6. Two girls, Agatha and Brionie, are roller skating inside a large empty building. The girls are modelled as particles. At time \(t = 0\), Agatha is at the point with position vector \(( 11 \mathbf { i } + 11 \mathbf { j } ) \mathrm { m }\) and Brionie is at the point with position vector \(( 7 \mathbf { i } + 16 \mathbf { j } ) \mathrm { m }\). The position vectors are given relative to the door, \(O\), and \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal perpendicular unit vectors. Agatha skates with constant velocity ( \(3 \mathbf { i } - \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\)
Brionie skates with constant velocity ( \(4 \mathbf { i } - 2 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\)

1. Find the position vector of Agatha at time \(t\) seconds. At time \(t = 6\) seconds, Agatha passes through the point \(P\).
2. Show that Brionie also passes through \(P\) and find the value of \(t\) when this occurs. At time \(t\) seconds, Agatha is at the point \(A\) and Brionie is at the point \(B\).
3. Show that \(\overrightarrow { A B } = [ ( t - 4 ) \mathbf { i } + ( 5 - t ) \mathbf { j } ] \mathrm { m }\)
4. Find the distance between the two girls when they are closest together. \includegraphics[max width=\textwidth, alt={}, center]{ca445c1e-078c-4a57-94df-de90f30f8efd-13_2255_50_314_34}
   

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7. A helicopter is hovering at rest above horizontal ground at the point \(H\). A parachutist steps out of the helicopter and immediately falls vertically and freely under gravity from rest for 2.5 s . His parachute then opens and causes him to immediately decelerate at a constant rate of \(3.9 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for \(T\) seconds ( \(T < 6\) ), until his speed is reduced to \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). He then moves with this constant speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) until he hits the ground. While he is decelerating, he falls a distance of 73.75 m . The total time between the instant when he leaves \(H\) and the instant when he hits the ground is 20 s . The parachutist is modelled as a particle.

1. Find the speed of the parachutist at the instant when his parachute opens.
2. Sketch a speed-time graph for the motion of the parachutist from the instant when he leaves \(H\) to the instant when he hits the ground.
3. Find the value of \(T\).
4. Find, to the nearest metre, the height of the point \(H\) above the ground.
   7. A helicopter is hovering at rest above horizontal ground at the point \(H\). A parachutist steps

8. \begin{figure}[h] \end{figure} Two particles, \(A\) and \(B\), have masses 2 kg and 4 kg respectively. The particles are connected by a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of a rough plane. The plane is inclined to the horizontal ground at an angle \(\alpha\) where \(\tan \alpha = \frac { 3 } { 4 }\). The particle \(A\) is held at rest on the plane at a distance \(d\) metres from the pulley. The particle \(B\) hangs freely at rest, vertically below the pulley, at a distance \(h\) metres above the ground, as shown in Figure 3. The part of the string between \(A\) and the pulley is parallel to a line of greatest slope of the plane. The coefficient of friction between \(A\) and the plane is \(\frac { 1 } { 4 }\) The system is released from rest with the string taut and \(B\) descends.

1. Find the tension in the string as \(B\) descends. On hitting the ground, \(B\) immediately comes to rest. Given that \(A\) comes to rest before reaching the pulley,
2. find, in terms of \(h\), the range of possible values of \(d\).
3. State one physical factor, other than air resistance, that could be taken into account to make the model described above more realistic.