Edexcel M1 (Mechanics 1) 2024 October

1. Particle \(A\) has mass \(4 m\) and particle \(B\) has mass \(3 m\).

The particles are moving in opposite directions along the same straight line on a smooth horizontal surface when they collide directly. Immediately before the collision, the speed of \(A\) is \(2 x\) and the speed of \(B\) is \(x\).
Immediately after the collision, the speed of \(A\) is \(y\) and the speed of \(B\) is \(5 y\).
The direction of motion of each particle is reversed as a result of the collision.

1. Show that \(y = \frac { 5 } { 11 } x\).
2. Find, in terms of \(m\) and \(x\), the magnitude of the impulse received by \(A\) in the collision.

2. \begin{figure}[h] \end{figure} A non-uniform beam \(A B\) has length 6 m and mass 50 kg . The beam rests horizontally on two supports at \(C\) and \(D\), where \(A C = 0.9 \mathrm {~m}\) and \(D B = 1.8 \mathrm {~m}\). A child of mass 25 kg stands on the beam at \(E\), where \(A E = E B = 3 \mathrm {~m}\), as shown in Figure 1. The beam is in equilibrium.
The magnitude of the normal reaction between the beam and the support at \(C\) is \(R _ { C }\) newtons. The magnitude of the normal reaction between the beam and the support at \(D\) is \(R _ { D }\) newtons. The beam is modelled as a rod and the child is modelled as a particle.
The centre of mass of the beam is between \(C\) and \(D\) and is a distance \(x\) metres from \(D\).
Given that \(2 R _ { D } = 3 R _ { C }\)

1. show that \(x = 1.38\) The child remains at \(E\) and a block of mass \(M \mathrm {~kg}\) is placed on the beam at \(B\).
   The block is modelled as a particle.
   Given that the beam is on the point of tilting,
2. find the value of \(M\).

1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors and position vectors are given relative to a fixed origin.]

A ship \(A\) is moving with constant velocity.
At 1 pm , the position vector of \(A\) is \(( 25 \mathbf { i } + 10 \mathbf { j } ) \mathrm { km }\).
At 3 pm , the position vector of \(A\) is \(( 55 \mathbf { i } + 34 \mathbf { j } ) \mathrm { km }\).
At time \(t\) hours after 1 pm , the position vector of \(A\) is \(\mathbf { r } _ { A } \mathrm {~km}\).

1. Show that \(\mathbf { r } _ { A } = ( 25 + 15 t ) \mathbf { i } + ( 10 + 12 t ) \mathbf { j }\) The speed of \(A\) is \(V \mathrm {~ms} ^ { - 1 }\)
2. Find the value of \(V\). A ship \(B\) is moving with constant velocity \(( 20 \mathbf { i } - 6 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\) At 1 pm , the position vector of \(B\) is \(( 35 \mathbf { i } + 51 \mathbf { j } ) \mathrm { km }\).
   At 2:30 pm, \(B\) passes through the point \(P\).
3. Show that \(A\) also passes through \(P\).

1. The points \(A\) and \(B\) lie on the same straight horizontal road.

Figure 2, on page 11, shows the speed-time graph of a cyclist \(P\), for his journey from \(A\) to \(B\).
At time \(t = 0 , P\) starts from rest at \(A\) and accelerates uniformly for 9 seconds until his speed is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) He then travels at constant speed \(V \mathrm {~ms} ^ { - 1 }\) When \(t = 42\), cyclist \(P\) passes \(B\).
Given that the distance \(A B\) is 120 m ,

1. show that \(V = 3.2\)
2. Find the acceleration of cyclist \(P\) between \(t = 0\) and \(t = 9\) Cyclist \(P\) continues to cycle along the road in the same direction at the same constant speed, \(V \mathrm {~ms} ^ { - 1 }\) When \(t = 6\), a second cyclist \(Q\) sets off from \(A\) and travels in the same direction as \(P\) along the same road. She accelerates for \(T\) seconds until her speed is \(3.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) She then travels at constant speed \(3.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Cyclist \(Q\) catches up with \(P\) when \(t = 54\)
3. On Figure 2, on page 11, sketch a speed-time graph showing the journeys of both cyclists, for the interval \(0 \leqslant t \leqslant 54\)
4. Find the value of \(T\) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2f2f89a6-cec4-444d-95d9-0112887d87eb-11_661_1509_292_278} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} A copy of Figure 2 is on page 13 if you need to redraw your answer to part (c). Only use this copy of Figure 2 if you need to redraw your answer to part (c). \includegraphics[max width=\textwidth, alt={}, center]{2f2f89a6-cec4-444d-95d9-0112887d87eb-13_666_1509_374_278} \section*{Copy of Figure 2}

5. \begin{figure}[h] \end{figure} Two particles, \(P\) and \(Q\), have masses 3 kg and 5 kg respectively. The particles are connected by a light inextensible string which passes over a small smooth fixed pulley. The particles are released from rest with the string taut and the hanging parts of the string vertical, as shown in Figure 3. Immediately after the particles are released from rest, \(P\) moves upwards with acceleration \(a \mathrm {~ms} ^ { - 2 }\) and the tension in the string is \(T\) newtons.

1. Write down an equation of motion for \(P\).
2. Find the value of \(T\). The total force acting on the pulley due to the string has magnitude \(F\) newtons.
3. Find the value of \(F\). Initially, \(Q\) is 10 m above horizontal ground and \(P\) is more than 2 m below the pulley.
   At the instant when \(Q\) has descended a distance of 2 m , the string breaks and \(Q\) falls to the ground.
4. Find the speed of \(Q\) at the instant it hits the ground.

6. \begin{figure}[h] \end{figure} A particle \(P\) of mass 5 kg lies on the surface of a rough plane.
The plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\) The particle is held in equilibrium by a horizontal force of magnitude \(H\) newtons, as shown in Figure 4. The horizontal force acts in a vertical plane containing a line of greatest slope of the inclined plane. The coefficient of friction between the particle and the plane is \(\frac { 1 } { 4 }\)

1. Find the smallest possible value of \(H\). The horizontal force is now removed, and \(P\) starts to slide down the slope.
   In the first \(T\) seconds after \(P\) is released from rest, \(P\) slides 1.5 m down the slope.
2. Find the value of \(T\).

7 At time \(t = 0\), a small ball \(A\) is projected vertically upwards with speed \(8 \mathrm {~ms} ^ { - 1 }\) from a fixed point on horizontal ground.
The ball hits the ground again for the first time at time \(t = T _ { 1 }\) seconds.
Ball \(A\) is modelled as a particle moving freely under gravity.

1. Show that \(T _ { 1 } = 1.63\) to 3 significant figures. After the first impact with the ground, \(A\) rebounds to a height of 2 m above the ground.
   Given that the mass of \(A\) is 0.1 kg ,
2. find the magnitude of the impulse received by \(A\) as a result of its first impact with the ground. At time \(t = 1\) second, another small ball \(B\) is projected vertically upwards from another point on the ground with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Ball \(B\) is modelled as a particle moving freely under gravity.
   At time \(t = T _ { 2 }\) seconds ( \(T _ { 2 } > 1\) ), \(A\) and \(B\) are at the same height above the ground for the first time.
3. Find the value of \(T _ { 2 }\)