AQA M1 (Mechanics 1) 2008 June

1 The diagram shows a velocity-time graph for a lift.
\includegraphics[max width=\textwidth, alt={}, center]{a381686b-0b1e-41ba-b88f-be1601e42098-2_337_917_552_557}

1. Find the distance travelled by the lift.
2. Find the acceleration of the lift during the first 4 seconds of the motion.
3. The lift is raised by a single vertical cable. The mass of the lift is 400 kg . Find the tension in the cable during the first 4 seconds of the motion.

2 The diagram shows three forces and the perpendicular unit vectors \(\mathbf { i }\) and \(\mathbf { j }\), which all lie in the same plane.
\includegraphics[max width=\textwidth, alt={}, center]{a381686b-0b1e-41ba-b88f-be1601e42098-2_415_398_1507_605}
\includegraphics[max width=\textwidth, alt={}, center]{a381686b-0b1e-41ba-b88f-be1601e42098-2_172_166_1567_1217}

1. Express the resultant of the three forces in terms of \(\mathbf { i }\) and \(\mathbf { j }\).
2. Find the magnitude of the resultant force.
3. Draw a diagram to show the direction of the resultant force, and find the angle that it makes with the unit vector \(\mathbf { i }\).

3 Two particles, \(A\) and \(B\), have masses 4 kg and 6 kg respectively. They are connected by a light inextensible string that passes over a smooth fixed peg. A second light inextensible string is attached to \(A\). The other end of this string is attached to the ground directly below \(A\). The system remains at rest, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{a381686b-0b1e-41ba-b88f-be1601e42098-3_457_711_523_845}

1. 1. Write down the tension in the string connecting \(A\) and \(B\).
   2. Find the tension in the string connecting \(A\) to the ground.
2. The string connecting particle \(A\) to the ground is cut. Find the acceleration of \(A\) after the string has been cut.

4 An aeroplane is travelling due north at \(180 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to the air. The air is moving north-west at \(50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the magnitude of the resultant velocity of the aeroplane.
2. Find the direction of the resultant velocity, giving your answer as a three-figure bearing to the nearest degree.

5 The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively. A helicopter moves horizontally with a constant acceleration of \(( - 0.4 \mathbf { i } + 0.5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). At time \(t = 0\), the helicopter is at the origin and has velocity \(20 \mathrm { i } \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. Write down an expression for the velocity of the helicopter at time \(t\) seconds.
2. Find the time when the helicopter is travelling due north.
3. Find an expression for the position vector of the helicopter at time \(t\) seconds.
4. When \(t = 100\) :
   1. show that the helicopter is due north of the origin;
   2. find the speed of the helicopter.

6 A block, of mass 5 kg , slides down a rough plane inclined at \(40 ^ { \circ }\) to the horizontal. When modelling the motion of the block, assume that there is no air resistance acting on it.

1. Draw and label a diagram to show the forces acting on the block.
2. Show that the magnitude of the normal reaction force acting on the block is 37.5 N , correct to three significant figures.
3. Given that the acceleration of the block is \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), find the coefficient of friction between the block and the plane.
4. In reality, air resistance does act on the block. State how this would change your value for the coefficient of friction and explain why.

7 A ball is hit by a bat so that, when it leaves the bat, its velocity is \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(35 ^ { \circ }\) above the horizontal. Assume that the ball is a particle and that its weight is the only force that acts on the ball after it has left the bat.

1. A simple model assumes that the ball is hit from the point \(A\) and lands for the first time at the point \(B\), which is at the same level as \(A\), as shown in the diagram.
   \includegraphics[max width=\textwidth, alt={}, center]{a381686b-0b1e-41ba-b88f-be1601e42098-4_321_1063_1370_484}
   1. Show that the time that it takes for the ball to travel from \(A\) to \(B\) is 4.68 seconds, correct to three significant figures.
   2. Find the horizontal distance from \(A\) to \(B\).
2. A revised model assumes that the ball is hit from the point \(C\), which is 1 metre above \(A\). The ball lands at the point \(D\), which is at the same level as \(A\), as shown in the diagram.
   \includegraphics[max width=\textwidth, alt={}, center]{a381686b-0b1e-41ba-b88f-be1601e42098-4_431_1177_2181_420} Find the time that it takes for the ball to travel from \(C\) to \(D\).

8 Two particles, \(A\) and \(B\), are travelling towards each other along a straight horizontal line.
Particle \(A\) has velocity \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and mass \(m \mathrm {~kg}\).
Particle \(B\) has velocity \(- 2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and mass 3 kg .
\includegraphics[max width=\textwidth, alt={}, center]{a381686b-0b1e-41ba-b88f-be1601e42098-5_220_1157_516_440} The particles collide.

1. If the particles move in opposite directions after the collision, each with speed \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the value of \(m\).
2. If the particles coalesce during the collision, forming a single particle which moves with speed \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the two possible values of \(m\).