AQA M1 (Mechanics 1) 2014 June

1 A car is travelling along a straight horizontal road. It is moving at \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it starts to accelerate. It accelerates at \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 12 seconds.

1. Find the speed of the car at the end of the 12 seconds.
2. Find the distance travelled during the 12 seconds.
3. The mass of the car is 1400 kg . A horizontal forward driving force of 1600 N acts on the car during the 12 seconds. Find the magnitude of the resistance force that acts on the car.
   [0pt] [3 marks]
   \includegraphics[max width=\textwidth, alt={}]{788534a5-abbb-4d6a-87b2-c54e859a128a-02_1513_1709_1192_153}

2 Three forces are in equilibrium in a vertical plane, as shown in the diagram. There is a vertical force of magnitude 40 N and a horizontal force of magnitude 60 N . The third force has magnitude \(F\) newtons and acts at an angle \(\theta\) above the horizontal.
\includegraphics[max width=\textwidth, alt={}, center]{788534a5-abbb-4d6a-87b2-c54e859a128a-04_490_894_456_571}

1. \(\quad\) Find \(F\).
2. \(\quad\) Find \(\theta\).

3 A skip, of mass 800 kg , is at rest on a rough horizontal surface. The coefficient of friction between the skip and the ground is 0.4 . A rope is attached to the skip and then the rope is pulled by a van so that the rope is horizontal while it is taut, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{788534a5-abbb-4d6a-87b2-c54e859a128a-06_237_1118_497_463} The mass of the van is 1700 kg . A constant horizontal forward driving force of magnitude \(P\) newtons acts on the van. The skip and the van accelerate at \(0.05 \mathrm {~ms} ^ { - 2 }\). Model both the van and the skip as particles connected by a light inextensible rope. Assume that there is no air resistance acting on the skip or on the van.

1. Find the speed of the van and the skip when they have moved 6 metres.
2. Draw a diagram to show the forces acting on the skip while it is accelerating.
3. Draw a diagram to show the forces acting on the van while it is accelerating. State one advantage of modelling the van as a particle when considering the vertical forces.
4. Find the magnitude of the friction force acting on the skip.
5. Find the tension in the rope.
6. \(\quad\) Find \(P\).
   \includegraphics[max width=\textwidth, alt={}]{788534a5-abbb-4d6a-87b2-c54e859a128a-06_771_1703_1932_155}

4 A boat is crossing a river, which has two parallel banks. The width of the river is 20 metres. The water in the river is flowing at a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The boat sets off from the point \(O\) on one bank. The point \(A\) is directly opposite \(O\) on the other bank. The velocity of the boat relative to the water is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(70 ^ { \circ }\) to the bank. The boat lands at the point \(B\) which is 3 metres from \(A\). The angle between the actual path of the boat and the bank is \(\alpha ^ { \circ }\). The river and the velocities are shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{788534a5-abbb-4d6a-87b2-c54e859a128a-10_490_1307_641_370}

1. Find the time that it takes for the boat to cross the river.
2. Find \(\alpha\).
3. \(\quad\) Find \(V\).

5 Two particles, \(A\) and \(B\), have masses of \(m\) and \(k m\) respectively, where \(k\) is a constant. The particles are moving on a smooth horizontal plane when they collide and coalesce to form a single particle. Just before the collision the velocities of \(A\) and \(B\) are \(( 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and \(( 6 \mathbf { i } - 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) respectively. Immediately after the collision the combined particle has velocity \(( 5.2 \mathbf { i } - 0.4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find \(k\).
[0pt] [5 marks]

6 A bullet is fired from a rifle at a target, which is at a distance of 420 metres from the rifle. The bullet leaves the rifle travelling at \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and at an angle of \(2 ^ { \circ }\) above the horizontal. The centre of the target, \(C\), is at the same horizontal level as the rifle. The bullet hits the target at the point \(A\), which is on a vertical line through \(C\). The bullet takes 1.8 seconds to reach the point \(A\).

1. Find \(V\), showing clearly how you obtain your answer.
2. Find the distance between \(A\) and \(C\).
3. State one assumption that you have made about the forces acting on the bullet.
   [0pt] [1 mark]

7 Two particles, \(A\) and \(B\), move on a horizontal surface with constant accelerations of \(- 0.4 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and \(0.2 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) respectively. At time \(t = 0\), particle \(A\) starts at the origin with velocity \(( 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). At time \(t = 0\), particle \(B\) starts at the point with position vector \(11.2 \mathbf { i }\) metres, with velocity \(( 0.4 \mathbf { i } + 0.6 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. Find the position vector of \(A , 10\) seconds after it leaves the origin.
   [0pt] [2 marks]
2. Show that the two particles collide, and find the position vector of the point where they collide.
   [0pt] [9 marks]
   \includegraphics[max width=\textwidth, alt={}]{788534a5-abbb-4d6a-87b2-c54e859a128a-16_1881_1707_822_153}
   \includegraphics[max width=\textwidth, alt={}]{788534a5-abbb-4d6a-87b2-c54e859a128a-17_2484_1707_221_153}

8 A crate, of mass 40 kg , is initially at rest on a rough slope inclined at \(30 ^ { \circ }\) to the horizontal, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{788534a5-abbb-4d6a-87b2-c54e859a128a-18_355_882_411_587} The coefficient of friction between the crate and the slope is \(\mu\).

1. Given that the crate is on the point of slipping down the slope, find \(\mu\).
2. A horizontal force of magnitude \(X\) newtons is now applied to the crate, as shown in the diagram.
   \includegraphics[max width=\textwidth, alt={}, center]{788534a5-abbb-4d6a-87b2-c54e859a128a-18_357_881_1208_575}
   1. Find the normal reaction on the crate in terms of \(X\).
   2. Given that the crate accelerates up the slope at \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), find \(X\).
      [0pt] [5 marks]
      \includegraphics[max width=\textwidth, alt={}]{788534a5-abbb-4d6a-87b2-c54e859a128a-19_2484_1707_221_153}