AQA M1 (Mechanics 1) 2006 June

1 A stone is dropped from a high bridge and falls vertically.

1. Find the distance that the stone falls during the first 4 seconds of its motion.
2. Find the average speed of the stone during the first 4 seconds of its motion.
3. State one modelling assumption that you have made about the forces acting on the stone during the motion.

2 A particle is in equilibrium under the action of four horizontal forces of magnitudes 5 newtons, 8 newtons, \(P\) newtons and \(Q\) newtons, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{cfe0bdbc-35e3-485f-a922-b652a72f4c95-2_355_357_1146_852}

1. Show that \(P = 9\).
2. Find the value of \(Q\).

3 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 \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) 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.

1. Show that the time taken for the third stage of the motion is 12.5 seconds.
2. Sketch a velocity-time graph for the car during the three stages of the motion.
3. Find the total distance travelled by the car during the motion.
4. State one criticism of the model of the motion.

4 A block is being pulled up a rough plane inclined at an angle of \(22 ^ { \circ }\) to the horizontal by a rope parallel to the plane, as shown in the diagram. The mass of the block is 0.7 kg , and the tension in the rope is \(T\) newtons.
\includegraphics[max width=\textwidth, alt={}, center]{cfe0bdbc-35e3-485f-a922-b652a72f4c95-3_264_460_1649_779}

1. Draw a diagram to show the forces acting on the block.
2. Show that the normal reaction force between the block and the plane has magnitude 6.36 newtons, correct to three significant figures.
3. The coefficient of friction between the block and the plane is 0.25 . Find the magnitude of the frictional force acting on the block during its motion.
4. The tension in the rope is 5.6 newtons. Find the acceleration of the block.

5 A small block \(P\) is attached to another small block \(Q\) by a light inextensible string. The block \(P\) rests on a rough horizontal surface and the string hangs over a smooth peg so that \(Q\) hangs freely, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{cfe0bdbc-35e3-485f-a922-b652a72f4c95-4_222_426_507_810} The block \(P\) has mass 0.4 kg and the coefficient of friction between \(P\) and the surface is 0.5 . The block \(Q\) has mass 0.3 kg . The system is released from rest and \(Q\) moves vertically downwards.

1. 1. Draw a diagram to show the forces acting on \(P\).
   2. Show that the frictional force between \(P\) and the surface has magnitude 1.96 newtons.
2. By forming an equation of motion for each block, show that the magnitude of the acceleration of each block is \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
3. Find the speed of the blocks after 3 seconds of motion.
4. After 3 seconds of motion, the string breaks. The blocks continue to move. Comment on how the speed of each block will change in the subsequent motion. For each block, give a reason for your answer.

6 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.

1. 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 \(B\).
2. 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 \(A C\).

7 A golf ball is struck from a point \(O\) with velocity \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(40 ^ { \circ }\) to the horizontal. The ball first hits the ground at a point \(P\), which is at a height \(h\) metres above the level of \(O\).
\includegraphics[max width=\textwidth, alt={}, center]{cfe0bdbc-35e3-485f-a922-b652a72f4c95-5_318_990_484_543} The horizontal distance between \(O\) and \(P\) is 57 metres.

1. Show that the time that the ball takes to travel from \(O\) to \(P\) is 3.10 seconds, correct to three significant figures.
2. Find the value of \(h\).
3. 1. Find the speed with which the ball hits the ground at \(P\).
   2. Find the angle between the direction of motion and the horizontal as the ball hits the ground at \(P\).

8 Two particles, \(A\) and \(B\), are moving on a smooth horizontal surface.
The particle \(A\) has mass \(m \mathrm {~kg}\) and is moving with velocity \(\left[ \begin{array} { r } 5
- 3 \end{array} \right] \mathrm { ms } ^ { - 1 }\). The particle \(B\) has mass 0.2 kg and is moving with velocity \(\left[ \begin{array} { l } 2
3 \end{array} \right] \mathrm { ms } ^ { - 1 }\).

1. Find, in terms of \(m\), an expression for the total momentum of the particles.
2. The particles \(A\) and \(B\) collide and form a single particle \(C\), which moves with velocity \(\left[ \begin{array} { c } k
   1 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(k\) is a constant.
   1. Show that \(m = 0.1\).
   2. Find the value of \(k\).