Edexcel M1 (Mechanics 1)

1. An aircraft moves along a straight horizontal runway with constant acceleration. It passes a point \(A\) on the runway with speed \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It then passes the point \(B\) on the runway with speed \(34 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The distance from \(A\) to \(B\) is 150 m .
   1. Find the acceleration of the aircraft.
   2. Find the time taken by the aircraft in moving from \(A\) to \(B\).
   3. Find, to 3 significant figures, the speed of the aircraft when it passes the point mid-way between \(A\) and \(B\).
   \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{ee3cbd24-55b1-4003-85bb-26d98f79a118-3_419_569_963_717} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} A particle has mass 2 kg . It is attached at \(B\) to the ends of two light inextensible strings \(A B\) and \(B C\). When the particle hangs in equilibrium, \(A B\) makes an angle of \(30 ^ { \circ }\) with the vertical, as shown in Fig. 1. The magnitude of the tension in \(B C\) is twice the magnitude of the tension in \(A B\).
2. Find, in degrees to one decimal place, the size of the angle that \(B C\) makes with the vertical.
   (4 marks)
3. Hence find, to 3 significant figures, the magnitude of the tension in \(A B\).

3. A racing car is travelling on a straight horizontal road. Its initial speed is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and it accelerates for 4 s to reach a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It then travels at a constant speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for a further 8 s . The total distance travelled by the car during this 12 s period is 600 m .

1. Sketch a speed-time graph to illustrate the motion of the car during this 12 s period.
2. Find the value of \(V\).
3. Find the acceleration of the car during the initial 4 s period.

4. \begin{figure}[h] \end{figure} A plank \(A B\) has length 4 m . It lies on a horizontal platform, with the end \(A\) lying on the platform and the end \(B\) projecting over the edge, as shown in Fig. 2. The edge of the platform is at the point \(C\). Jack and Jill are experimenting with the plank. Jack has mass 40 kg and Jill has mass 25 kg . They discover that, if Jack stands at \(B\) and Jill stands at \(A\) and \(B C = 1.6 \mathrm {~m}\), the plank is in equilibrium and on the point of tilting about \(C\). By modelling the plank as a uniform rod, and Jack and Jill as particles,

1. find the mass of the plank. They now alter the position of the plank in relation to the platform so that, when Jill stands at \(B\) and Jack stands at \(A\), the plank is again in equilibrium and on the point of tilting about \(C\).
2. Find the distance \(B C\) in this position.
3. State how you have used the modelling assumptions that
   1. the plank is uniform,
   2. the plank is a rod,
   3. Jack and Jill are particles.

5. A post is driven into the ground by means of a blow from a pile-driver. The pile-driver falls from rest from a height of 1.6 m above the top of the post.

1. Show that the speed of the pile-driver just before it hits the post is \(5.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). (2 marks) The post has mass 6 kg and the pile-driver has mass 78 kg . When the pile-driver hits the top of the post, it is assumed that the there is no rebound and that both then move together with the same speed.
2. Find the speed of the pile-driver and the post immediately after the pile-driver has hit the post. The post is brought to rest by the action of a resistive force from the ground acting for 0.06 s .
   By modelling this force as constant throughout this time,
3. find the magnitude of the resistive force,
4. find, to 2 significant figures, the distance travelled by the post and the pile-driver before they come to rest.
   (4 marks)

6. [In this question, the horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed due East and North respectively.] A coastguard station \(O\) monitors the movements of ships in a channel. At noon, the station's radar records two ships moving with constant speed. Ship \(A\) is at the point with position vector \(( - 5 \mathbf { i } + 10 \mathbf { j } ) \mathrm { km }\) relative to \(O\) and has velocity \(( 2 \mathbf { i } + 2 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\). Ship \(B\) is at the point with position vector \(( 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { km }\) and has velocity \(( - 2 \mathbf { i } + 5 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\).

1. Given that the two ships maintain these velocities, show that they collide. The coast guard radios ship \(A\) and orders it to reduce its speed to move with velocity \(( \mathbf { i } + \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\). Given that \(A\) obeys this order and maintains this new constant velocity,
2. find an expression for the vector \(\overrightarrow { A B }\) at time \(t\) hours after noon.
3. find, to 3 significant figures, the distance between \(A\) and \(B\) at 1400 hours,
4. find the time at which \(B\) will be due north of \(A\).

7. \begin{figure}[h] \end{figure} A small parcel of mass 2 kg moves on a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The parcel is pulled up a line of greatest slope of the plane by means of a light rope which it attached to it. The rope makes an angle of \(30 ^ { \circ }\) with the plane, as shown in Fig. 3. The coefficient of friction between the parcel and the plane is 0.4 Given that the tension in the rope is 24 N ,

1. find, to 2 significant figures, the acceleration of the parcel. The rope now breaks. The parcel slows down and comes to rest.
2. Show that, when the parcel comes to this position of rest, it immediately starts to move down the plane again.
3. Find, to 2 significant figures, the acceleration of the parcel as it moves down the plane after it has come to this position of instantaneous rest.