AQA M1 (Mechanics 1) 2011 January

1 A trolley, of mass 5 kg , is moving in a straight line on a smooth horizontal surface. It has a velocity of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it collides with a stationary trolley, of mass \(m \mathrm {~kg}\). Immediately after the collision, the trolleys move together with velocity \(2.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find \(m\).
(3 marks)

2 The graph shows how the velocity of a train varies as it moves along a straight railway line.
\includegraphics[max width=\textwidth, alt={}, center]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-04_574_1595_402_203}

1. Find the total distance travelled by the train.
2. Find the average speed of the train.
3. Find the acceleration of the train during the first 10 seconds of its motion.
4. The mass of the train is 200 tonnes. Find the magnitude of the resultant force acting on the train during the first 10 seconds of its motion.

3 A car, of mass 1200 kg , tows a caravan, of mass 1000 kg , along a straight horizontal road. The caravan is attached to the car by a horizontal tow bar, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-06_277_901_484_584} Assume that a constant resistance force of magnitude 200 newtons acts on the car and a constant resistance force of magnitude 300 newtons acts on the caravan. A constant driving force of magnitude \(P\) newtons acts on the car in the direction of motion. The car and caravan accelerate at \(0.8 \mathrm {~ms} ^ { - 2 }\).

1. 1. Find \(P\).
   2. Find the magnitude of the force in the tow bar that connects the car to the caravan.
2. 1. Find the time that it takes for the speed of the car and caravan to increase from \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   2. Find the distance that they travel in this time.
3. Explain why the assumption that the resistance forces are constant is unrealistic.
   (1 mark)

4 A canoe is paddled across a river which has a width of 20 metres. The canoe moves from the point \(X\) on one bank of the river to the point \(Y\) on the other bank, so that its path is a straight line at an angle \(\alpha\) to the banks. The velocity of the canoe relative to the water is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) perpendicular to the banks. The water flows at \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) parallel to the banks.
\includegraphics[max width=\textwidth, alt={}, center]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-10_469_1333_543_374} Model the canoe as a particle.

1. Find the magnitude of the resultant velocity of the canoe.
2. Find the angle \(\alpha\).
3. Find the time that it takes for the canoe to travel from \(X\) to \(Y\).
   \includegraphics[max width=\textwidth, alt={}]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-11_2486_1714_221_153}

5 A particle moves with constant acceleration \(( - 0.4 \mathbf { i } + 0.2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). Initially, it has velocity \(( 4 \mathbf { i } + 0.5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.

1. Find an expression for the velocity of the particle at time \(t\) seconds.
2. 1. Find the velocity of the particle when \(t = 22.5\).
   2. State the direction in which the particle is travelling at this time.
3. Find the time when the speed of the particle is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

6 Two particles, \(A\) and \(B\), are connected by a light inextensible string which passes over a smooth peg. Particle \(A\) has mass 2 kg and particle \(B\) has mass 4 kg . Particle \(A\) hangs freely with the string vertical. Particle \(B\) is at rest in equilibrium on a rough horizontal surface with the string at an angle of \(30 ^ { \circ }\) to the vertical. The particles, peg and string are shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-14_419_953_571_541}

1. By considering particle \(A\), find the tension in the string.
2. Draw a diagram to show the forces acting on particle \(B\).
3. Show that the magnitude of the normal reaction force acting on particle \(B\) is 22.2 newtons, correct to three significant figures.
4. Find the least possible value of the coefficient of friction between particle \(B\) and the surface.
   \includegraphics[max width=\textwidth, alt={}]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-16_2486_1714_221_153}

7 An arrow is fired from a point at a height of 1.5 metres above horizontal ground. It has an initial velocity of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal. The arrow hits a target at a height of 1 metre above horizontal ground. The path of the arrow is shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-18_341_1260_550_390} Model the arrow as a particle.

1. Show that the time taken for the arrow to travel to the target is 1.30 seconds, correct to three significant figures.
2. Find the horizontal distance between the point where the arrow is fired and the target.
3. Find the speed of the arrow when it hits the target.
4. Find the angle between the velocity of the arrow and the horizontal when the arrow hits the target.
5. State one assumption that you have made about the forces acting on the arrow.
   (1 mark)
   \includegraphics[max width=\textwidth, alt={}]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-19_2486_1714_221_153}
   \includegraphics[max width=\textwidth, alt={}]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-20_2486_1714_221_153}

8 A van, of mass 2000 kg , is towed up a slope inclined at \(5 ^ { \circ }\) to the horizontal. The tow rope is at an angle of \(12 ^ { \circ }\) to the slope. The motion of the van is opposed by a resistance force of magnitude 500 newtons. The van is accelerating up the slope at \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
\includegraphics[max width=\textwidth, alt={}, center]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-22_269_991_513_529} Model the van as a particle.

1. Draw a diagram to show the forces acting on the van.
2. Show that the tension in the tow rope is 3480 newtons, correct to three significant figures.