AQA M1 (Mechanics 1)

1 A particle \(A\) moves across a smooth horizontal surface in a straight line. The particle \(A\) has mass 2 kg and speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A particle \(B\), which has mass 3 kg , is at rest on the surface. The particle \(A\) collides with the particle \(B\).
\includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-003_147_506_644_733}

1. If, after the collision, \(A\) is at rest and \(B\) moves away from \(A\), find the speed of \(B\).
2. If, after the collision, \(A\) and \(B\) move away from each other with speeds \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(4 v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively, as shown in the diagram below, find the value of \(v\).
   \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-003_138_506_1144_730}

4 Water flows in a constant direction at a constant speed of \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A boat travels in the water at a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to the water.

1. The direction in which the boat travels relative to the water is perpendicular to the direction of motion of the water. The resultant velocity of the boat is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(74 ^ { \circ }\) to the direction of motion of the water, as shown in the diagram. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-004_120_164_662_488} \captionsetup{labelformat=empty} \caption{Velocity of the water}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-004_126_186_667_890} \captionsetup{labelformat=empty} \caption{Velocity of the boat relative to the water}
   \end{figure}
   1. Find \(V\).
   2. Show that \(u = 3.44\), correct to three significant figures.
2. The boat changes course so that it travels relative to the water at an angle of \(45 ^ { \circ }\) to the direction of motion of the water. The resultant velocity of the boat is now of magnitude \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The velocity of the water is unchanged, as shown in the diagram below. $$\xrightarrow { 3.44 \mathrm {~m} \mathrm {~s} ^ { - 1 } }$$
   \includegraphics[max width=\textwidth, alt={}]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-004_132_273_1493_895}
   Velocity of the boat relative to the water
   \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-004_232_355_1498_1384} Find the value of \(v\).
   (4 marks)

5 A golf ball is projected from a point \(O\) with initial velocity \(V\) at an angle \(\alpha\) to the horizontal. The ball first hits the ground at a point \(A\) which is at the same horizontal level as \(O\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-005_227_602_484_735} It is given that \(V \cos \alpha = 6 u\) and \(V \sin \alpha = 2.5 u\).

1. Show that the time taken for the ball to travel from \(O\) to \(A\) is \(\frac { 5 u } { g }\).
2. Find, in terms of \(g\) and \(u\), the distance \(O A\).
3. Find \(V\), in terms of \(u\).
4. State, in terms of \(u\), the least speed of the ball during its flight from \(O\) to \(A\).

6 A van moves from rest on a straight horizontal road.

1. In a simple model, the first 30 seconds of the motion are represented by three separate stages, each lasting 10 seconds and each with a constant acceleration. During the first stage, the van accelerates from rest to a velocity of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   During the second stage, the van accelerates from \(4 \mathrm {~ms} ^ { - 1 }\) to \(12 \mathrm {~ms} ^ { - 1 }\).
   During the third stage, the van accelerates from \(12 \mathrm {~ms} ^ { - 1 }\) to \(16 \mathrm {~ms} ^ { - 1 }\).
   1. Sketch a velocity-time graph to represent the motion of the van during the first 30 seconds of its motion.
   2. Find the total distance that the van travels during the 30 seconds.
   3. Find the average speed of the van during the 30 seconds.
   4. Find the greatest acceleration of the van during the 30 seconds.
2. In another model of the 30 seconds of the motion, the acceleration of the van is assumed to vary during the first and third stages of the motion, but to be constant during the second stage, as shown in the velocity-time graph below.
   \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-006_554_1138_1432_539} The velocity of the van takes the same values at the beginning and the end of each stage of the motion as in part (a).
   1. State, with a reason, whether the distance travelled by the van during the first 10 seconds of the motion in this model is greater or less than the distance travelled during the same time interval in the model in part (a).
   2. Give one reason why this model represents the motion of the van more realistically than the model in part (a).

7 A builder ties two identical buckets, \(P\) and \(Q\), to the ends of a light inextensible rope. He hangs the rope over a smooth beam so that the buckets hang in equilibrium, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-007_360_296_502_904} The buckets are each of mass 0.6 kg .

1. 1. State the magnitude of the tension in the rope.
   2. State the magnitude and direction of the force exerted on the beam by the rope.
2. The bucket \(Q\) is held at rest while a stone, of mass 0.2 kg , is placed inside it. The system is then released from rest and, in the subsequent motion, bucket \(Q\) moves vertically downwards with the stone inside.
   1. By forming an equation of motion for each bucket, show that the magnitude of the tension in the rope during the motion is 6.72 newtons, correct to three significant figures.
   2. State the magnitude of the force exerted on the beam by the rope while the motion takes place.

8 A rough slope is inclined at an angle of \(25 ^ { \circ }\) to the horizontal. A box of weight 80 newtons is on the slope. A rope is attached to the box and is parallel to the slope. The tension in the rope is of magnitude \(T\) newtons. The diagram shows the slope, the box and the rope.
\includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-008_307_469_500_840}

1. The box is held in equilibrium by the rope.
   1. Show that the normal reaction force between the box and the slope is 72.5 newtons, correct to three significant figures.
   2. The coefficient of friction between the box and the slope is 0.32 . Find the magnitude of the maximum value of the frictional force which can act on the box.
   3. Find the least possible tension in the rope to prevent the box from moving down the slope.
   4. Find the greatest possible tension in the rope.
   5. Show that the mass of the box is approximately 8.16 kg .
2. The rope is now released and the box slides down the slope. Find the acceleration of the box. General Certificate of Education
   June 2006
   Advanced Subsidiary Examination ASSESSMENT and
   REALIFIEATIONS
   ALLIANCE Tuesday 6 June 20061.30 pm to 2.45 pm \section*{For this paper you must have:}
   • an 8-page answer book
   • the blue AQA booklet of formulae and statistical tables
   You may use a graphics calculator. Time allowed: 1 hour 15 minutes \section*{Instructions}
   • Use blue or black ink or ball-point pen. Pencil should only be used for drawing.
   • Write the information required on the front of your answer book. The Examining Body for this paper is AQA. The Paper Reference is MM1A/W.
   • Answer all questions.
   • Show all necessary working; otherwise marks for method may be lost.
   • The final answer to questions requiring the use of calculators should be given to three significant figures, unless stated otherwise.
   • Take \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), unless stated otherwise.
   \section*{Information}
   • The maximum mark for this paper is 60 .
   • The marks for questions are shown in brackets.
   • Unit Mechanics 1A has a written paper and coursework.
   \section*{Advice}
   • Unless stated otherwise, you may quote formulae, without proof, from the booklet.
   Answer all questions. 1 A small stone is dropped from a high bridge and falls vertically.
3. Find the distance that the stone falls during the first 4 seconds of its motion. (3 marks)
4. Find the speed of the stone when it has been falling for 4 seconds. 2 A car travels along a straight horizontal road. The motion of the car can be modelled as three separate stages. During the first stage, the car accelerates uniformly from rest to a velocity of \(10 \mathrm {~ms} ^ { - 1 }\) in 6 seconds. During the second stage, the car travels with a constant velocity of \(10 \mathrm {~ms} ^ { - 1 }\) for a further 4 seconds. During the third stage of the motion, the car travels with a uniform retardation of magnitude \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it comes to rest.
5. Show that the time taken for the third stage of the motion is 12.5 seconds.
6. Sketch a velocity-time graph for the car during the three stages of the motion.
7. Find the total distance travelled by the car during the motion. 3 A stone rests in equilibrium on a rough plane inclined at an angle of \(16 ^ { \circ }\) to the horizontal, as shown in the diagram. The mass of the stone is 0.5 kg .
   \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-010_222_528_1978_731}
8. Draw a diagram to show the forces acting on the stone.
9. Show that the magnitude of the frictional force acting on the stone is 1.35 newtons, correct to three significant figures.
10. Find the magnitude of the normal reaction force between the stone and the plane.
11. Hence find an inequality for the value of \(\mu\), the coefficient of friction between the stone and the plane. 4 A block \(P\) is attached to a can \(Q\) by a light inextensible string. The string hangs over a smooth peg so that \(P\) and \(Q\) hang freely, as shown in the diagram.
    \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-011_246_259_470_886} The block \(P\) and the can \(Q\) each has mass 0.2 kg . The can \(Q\) contains a small stone of mass 0.1 kg . The system is released from rest and the can \(Q\) and the stone move vertically downwards.
12. By forming two equations of motion, show that the magnitude of the acceleration of \(P\) and \(Q\) is \(1.96 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
13. Find the magnitude of the reaction force between the can and the stone. 5 The points \(A\) and \(B\) have position vectors \(( 3 \mathbf { i } + 2 \mathbf { j } )\) metres and \(( 6 \mathbf { i } - 4 \mathbf { j } )\) metres respectively. The vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a horizontal plane.
14. A particle moves from \(A\) to \(B\) with constant velocity \(( \mathbf { i } - 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). Calculate the time that the particle takes to move from \(A\) to \(B\).
15. The particle then moves from \(B\) to a point \(C\) with a constant acceleration of \(2 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 2 }\). It takes 4 seconds to move from \(B\) to \(C\).
    1. Find the position vector of \(C\).
    2. Find the distance \(A C\).