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OCR M2 2010 June Q6
17 marks Standard +0.3
A particle \(A\) of mass \(2m\) is moving with speed \(u\) on a smooth horizontal surface when it collides with a stationary particle \(B\) of mass \(m\). After the collision the speed of \(A\) is \(v\), the speed of \(B\) is \(3v\) and the particles move in the same direction.
  1. Find \(v\) in terms of \(u\). [3]
  2. Show that the coefficient of restitution between \(A\) and \(B\) is \(\frac{1}{3}\). [2]
\(B\) subsequently hits a vertical wall which is perpendicular to the direction of motion. As a result of the impact, \(B\) loses \(\frac{3}{4}\) of its kinetic energy.
  1. Show that the speed of \(B\) after hitting the wall is \(\frac{3}{4}u\). [4]
  2. \(B\) then hits \(A\). Calculate the speeds of \(A\) and \(B\), in terms of \(u\), after this collision and state their directions of motion. [8]
OCR M2 2010 June Q7
13 marks Standard +0.8
A small ball of mass 0.2 kg is projected with speed \(11 \text{ ms}^{-1}\) up a line of greatest slope of a roof from a point \(A\) at the bottom of the roof. The ball remains in contact with the roof and moves up the line of greatest slope to the top of the roof at \(B\). The roof is rough and the coefficient of friction is \(\frac{1}{4}\). The distance \(AB\) is 5 m and \(AB\) is inclined at \(30°\) to the horizontal (see diagram).
  1. Show that the speed of the ball when it reaches \(B\) is \(5.44 \text{ ms}^{-1}\), correct to 2 decimal places. [6]
The ball leaves the roof at \(B\) and moves freely under gravity. The point \(C\) is at the lower edge of the roof. The distance \(BC\) is 5 m and \(BC\) is inclined at \(30°\) to the horizontal.
  1. Determine whether or not the ball hits the roof between \(B\) and \(C\). [7]
OCR M2 2016 June Q1
6 marks Moderate -0.3
A car of mass 1400 kg is travelling on a straight horizontal road against a constant resistance to motion of 600 N. At a certain instant the car is accelerating at \(0.3 \text{ m s}^{-2}\) and the engine of the car is working at a rate of 23 kW.
  1. Find the speed of the car at this instant. [3]
Subsequently the car moves up a hill inclined at \(10°\) to the horizontal at a steady speed of \(12 \text{ m s}^{-1}\). The resistance to motion is still a constant 600 N.
  1. Calculate the power of the car's engine as it moves up the hill. [3]
OCR M2 2016 June Q2
7 marks Standard +0.3
\(A\) and \(B\) are two points on a line of greatest slope of a plane inclined at \(55°\) to the horizontal. \(A\) is below the level of \(B\) and \(AB = 4\) m. A particle \(P\) of mass 2.5 kg is projected up the plane from \(A\) towards \(B\) and the speed of \(P\) at \(B\) is \(6.7 \text{ m s}^{-1}\). The coefficient of friction between the plane and \(P\) is 0.15. Find
  1. the work done against the frictional force as \(P\) moves from \(A\) to \(B\), [3]
  2. the initial speed of \(P\) at \(A\). [4]
OCR M2 2016 June Q3
12 marks Standard +0.3
\includegraphics{figure_1} A uniform lamina \(ABDC\) is bounded by two semicircular arcs \(AB\) and \(CD\), each with centre \(O\) and of radii \(3a\) and \(a\) respectively, and two straight edges, \(AC\) and \(DB\), which lie on the line \(AOB\) (see Fig. 1).
  1. Show that the distance of the centre of mass of the lamina from \(O\) is \(\frac{13a}{3\pi}\). [5]
\includegraphics{figure_2} The lamina has mass 3 kg and is freely pivoted to a fixed point at \(A\). The lamina is held in equilibrium with \(AB\) vertical by means of a light string attached to \(B\). The string lies in the same plane as the lamina and is at an angle of \(40°\) below the horizontal (see Fig. 2).
  1. Calculate the tension in the string. [3]
  2. Find the direction of the force acting on the lamina at \(A\). [4]
OCR M2 2016 June Q4
9 marks Standard +0.8
A smooth solid cone of semi-vertical angle \(60°\) is fixed to the ground with its axis vertical. A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point vertically above the vertex of the cone. \(P\) rotates in a horizontal circle on the surface of the cone with constant angular velocity \(\omega\). The string is inclined to the downward vertical at an angle of \(30°\) (see diagram).
  1. Show that the magnitude of the contact force between the cone and the particle is \(\frac{1}{4}m(2\sqrt{3}g - 3a\omega^2)\). [6]
  2. Given that \(a = 0.5\) m and \(m = 3.5\) kg, find, in either order, the greatest speed for which the particle remains in contact with the cone and the corresponding tension in the string. [3]
OCR M2 2016 June Q5
11 marks Standard +0.3
A uniform ladder \(AB\), of weight \(W\) and length \(2a\), rests with the end \(A\) in contact with rough horizontal ground and the end \(B\) resting against a smooth vertical wall. The ladder is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{12}{13}\). A man of weight \(6W\) is standing on the ladder at a distance \(x\) from \(A\) and the system is in equilibrium.
  1. Show that the magnitude of the frictional force exerted by the ground on the ladder is \(\frac{5W}{24}\left(1 + \frac{6x}{a}\right)\). [5]
The coefficient of friction between the ladder and the ground is \(\frac{1}{3}\).
  1. Find, in terms of \(a\), the greatest value of \(x\) for which the system is in equilibrium. [3]
The bottom of the ladder \(A\) is moved closer to the wall so that the ladder is now inclined at an angle \(\alpha\) to the horizontal. The man of weight \(6W\) can now stand at the top of the ladder \(B\) without the ladder slipping.
  1. Find the least possible value of \(\tan \alpha\). [3]
OCR M2 2016 June Q6
10 marks Standard +0.8
The masses of two particles \(A\) and \(B\) are 4 kg and 3 kg respectively. The particles are moving towards each other along a straight line on a smooth horizontal surface. \(A\) has speed \(8 \text{ m s}^{-1}\) and \(B\) has speed \(10 \text{ m s}^{-1}\) before they collide. The kinetic energy lost due to the collision is 121.5 J.
  1. Find the speed and direction of motion of each particle after the collision. [8]
  2. Find the coefficient of restitution between \(A\) and \(B\). [2]
OCR M2 2016 June Q7
17 marks Challenging +1.8
A particle \(P\) is projected with speed \(32 \text{ m s}^{-1}\) at an angle of elevation \(\alpha\), where \(\sin \alpha = \frac{3}{4}\), from a point \(A\) on horizontal ground. At the same instant a particle \(Q\) is projected with speed \(20 \text{ m s}^{-1}\) at an angle of elevation \(\beta\), where \(\sin \beta = \frac{24}{25}\), from a point \(B\) on the same horizontal ground. The particles move freely under gravity in the same vertical plane and collide with each other at the point \(C\) at the instant when they are travelling horizontally (see diagram).
  1. Calculate the height of \(C\) above the ground and the distance \(AB\). [4]
Immediately after the collision \(P\) falls vertically. \(P\) hits the ground and rebounds vertically upwards, coming to instantaneous rest at a height 5 m above the ground.
  1. Given that the mass of \(P\) is 3 kg, find the magnitude and direction of the impulse exerted on \(P\) by the ground. [4]
The coefficient of restitution between the two particles is \(\frac{1}{2}\).
  1. Find the distance of \(Q\) from \(C\) at the instant when \(Q\) is travelling in a direction of \(25°\) below the horizontal. [9]
OCR MEI M2 2007 January Q1
17 marks Moderate -0.3
A sledge and a child sitting on it have a combined mass of 29.5 kg. The sledge slides on horizontal ice with negligible resistance to its movement.
  1. While at rest, the sledge is hit directly from behind by a ball of mass 0.5 kg travelling horizontally at \(10 \text{ m s}^{-1}\). The coefficient of restitution in the collision is 0.8. After the impact the speeds of the sledge and the ball are \(V_1 \text{ m s}^{-1}\) and \(V_2 \text{ m s}^{-1}\) respectively. Calculate \(V_1\) and \(V_2\) and state the direction in which the ball is travelling after the impact. [7]
  2. While at rest, the sledge is hit directly from behind by a snowball of mass 0.5 kg travelling horizontally at \(10 \text{ m s}^{-1}\). The snowball sticks to the sledge.
    1. Calculate the velocity with which the combined sledge and snowball start to move. [3]
    2. The child scoops up the 0.5 kg of snow and drops it over the back of the sledge. What happens to the velocity of the sledge? Give a reason for your answer. [2]
  3. In another situation, the sledge is travelling over the ice at \(2 \text{ m s}^{-1}\) with 10.5 kg of snow on it (giving a total mass of 40 kg). The child throws a snowball of mass 0.5 kg from the sledge, parallel to the ground and in the positive direction of the motion of the sledge. Immediately after the snowball is thrown, the sledge has a speed of \(V \text{ m s}^{-1}\) and the snowball and sledge are separating at a speed of \(10 \text{ m s}^{-1}\). Draw a diagram showing the velocities of the sledge and snowball before and after the snowball is thrown. Calculate \(V\). [5]
OCR MEI M2 2007 January Q2
20 marks Standard +0.8
\includegraphics{figure_2} Fig. 2 shows a framework in a vertical plane made from the equal, light, rigid rods AB, BC, AD, BD, BE, CE and DE. [The triangles ABD, BDE and BCE are all equilateral.] The rods AB, BC and DE are horizontal. The rods are freely pin-jointed to each other at A, B, C, D and E. The pin-joint at A is also fixed to an inclined plane. The plane is smooth and parallel to the rod AD. The pin-joint at D rests on this plane. The following external forces act on the framework: a vertical load of \(LN\) at C; the normal reaction force \(RN\) of the plane on the framework at D; the horizontal and vertical forces \(XN\) and \(YN\), respectively, acting at A.
  1. Write down equations for the horizontal and vertical equilibrium of the framework. [3]
  2. By considering moments, find the relationship between \(R\) and \(L\). Hence show that \(X = \sqrt{3}L\) and \(Y = 0\). [4]
  3. Draw a diagram showing all the forces acting on the pin-joints, including the forces internal to the rods. [2]
  4. Show that the internal force in the rod AD is zero. [2]
  5. Find the forces internal to AB, CE and BC in terms of \(L\) and state whether each is a tension or a thrust (compression). [You may leave your answers in surd form.] [7]
  6. Without calculating their values in terms of \(L\), show that the forces internal to the rods BD and BE have equal magnitude but one is a tension and the other a thrust. [2]
OCR MEI M2 2007 January Q3
18 marks Standard +0.8
A box is to be assembled in the shape of the cuboid shown in Fig. 3.1. The lengths are in centimetres. All the faces are made of the same uniform, rigid and thin material. All coordinates refer to the axes shown in this figure. \includegraphics{figure_3.1}
  1. The four vertical faces OAED, ABFE, FGCB and CODG are assembled first to make an open box without a base or a top. Write down the coordinates of the centre of mass of this open box. [1]
The base OABC is added to the vertical faces.
  1. Write down the \(x\)- and \(y\)-coordinates of the centre of mass of the box now. Show that the \(z\)-coordinate is now 1.875. [5]
The top face FGDE is now added. This is a lid hinged to the rest of the box along the line FG. The lid is open so that it hangs in a vertical plane touching the face FGCB.
  1. Show that the coordinates of the centre of mass of the box in this situation are \((10, 2.4, 2.1)\). [6]
[This question is continued on the facing page.] The box, with the lid still touching face FGCB, is now put on a sloping plane with the edge OA horizontal and the base inclined at \(30°\) to the horizontal, as shown in Fig. 3.2. \includegraphics{figure_3.2} The weight of the box is 40 N. A force \(P\) N acts parallel to the plane and is applied to the mid-point of FG at \(90°\) to FG. This force tends to push the box down the plane. The box does not slip and is on the point of toppling about the edge AO.
  1. Show that the clockwise moment of the weight of the box about the edge AO is about 0.411 Nm. [4]
  2. Calculate the value of \(P\). [2]
OCR MEI M2 2007 January Q4
17 marks Standard +0.3
Some tiles on a roof are being replaced. Each tile has a mass of 2 kg and the coefficient of friction between it and the existing roof is 0.75. The roof is at \(30°\) to the horizontal and the bottom of the roof is 6 m above horizontal ground, as shown in Fig. 4. \includegraphics{figure_4}
  1. Calculate the limiting frictional force between a tile and the roof. A tile is placed on the roof. Does it slide? (Your answer should be supported by a calculation.) [5]
  2. The tiles are raised 6 m from the ground, the only work done being against gravity. They are then slid 4 m up the roof and placed at the point A shown in Fig. 4.
    1. Show that each tile gains 156.8 J of gravitational potential energy. [3]
    2. Calculate the work done against friction per tile. [2]
    3. What average power is required to raise 10 tiles per minute from the ground to A? [2]
  3. A tile is kicked from A directly down the roof. When the tile is at B, \(x\) m from the edge of the roof, its speed is \(4 \text{ m s}^{-1}\). It subsequently hits the ground travelling at \(9 \text{ m s}^{-1}\). In the motion of the tile from B to the ground, the work done against sliding and other resistances is 90 J. Use an energy method to find \(x\). [5]
OCR MEI M2 2008 January Q1
19 marks Moderate -0.3
  1. A battering-ram consists of a wooden beam fixed to a trolley. The battering-ram runs along horizontal ground and collides directly with a vertical wall, as shown in Fig. 1.1. The battering-ram has a mass of 4000 kg. \includegraphics{figure_1} Initially the battering-ram is at rest. Some men push it for 8 seconds and let go just as it is about to hit the wall. While the battering-ram is being pushed, the constant overall force on it in the direction of its motion is 1500 N.
    1. At what speed does the battering-ram hit the wall? [3]
    The battering-ram hits a loose stone block of mass 500 kg in the wall. Linear momentum is conserved and the coefficient of restitution in the impact is 0.2.
    1. Calculate the speeds of the stone block and of the battering-ram immediately after the impact. [6]
    2. Calculate the energy lost in the impact. [3]
  2. Small objects A and B are sliding on smooth, horizontal ice. Object A has mass 4 kg and speed 18 m s\(^{-1}\) in the \(\mathbf{i}\) direction. B has mass 8 kg and speed 9 m s\(^{-1}\) in the direction shown in Fig. 1.2, where \(\mathbf{i}\) and \(\mathbf{j}\) are the standard unit vectors. \includegraphics{figure_2}
    1. Write down the linear momentum of A and show that the linear momentum of B is \((36\mathbf{i} + 36\sqrt{3}\mathbf{j})\) N s. [2]
    After the objects meet they stick together (coalesce) and move with a common velocity of \((u\mathbf{i} + v\mathbf{j})\) m s\(^{-1}\).
    1. Calculate \(u\) and \(v\). [3]
    2. Find the angle between the direction of motion of the combined object and the \(\mathbf{i}\) direction. Make your method clear. [2]
OCR MEI M2 2008 January Q2
17 marks Moderate -0.3
A cyclist and her bicycle have a combined mass of 80 kg.
  1. Initially, the cyclist accelerates from rest to 3 m s\(^{-1}\) against negligible resistances along a horizontal road.
    1. How much energy is gained by the cyclist and bicycle? [2]
    2. The cyclist travels 12 m during this acceleration. What is the average driving force on the bicycle? [2]
  2. While exerting no driving force, the cyclist free-wheels down a hill. Her speed increases from 4 m s\(^{-1}\) to 10 m s\(^{-1}\). During this motion, the total work done against friction is 1600 J and the drop in vertical height is \(h\) m. Without assuming that the hill is uniform in either its angle or roughness, calculate \(h\). [5]
  3. The cyclist reaches another horizontal stretch of road and there is now a constant resistance to motion of 40 N.
    1. When the power of the driving force on the bicycle is a constant 200 W, what constant speed can the cyclist maintain? [3]
    2. Find the power of the driving force on the bicycle when travelling at a speed of 0.5 m s\(^{-1}\) with an acceleration of 2 m s\(^{-2}\). [5]
OCR MEI M2 2008 January Q3
18 marks Standard +0.3
A lamina is made from uniform material in the shape shown in Fig. 3.1. BCJA, DZOJ, ZEIO and FGHI are all rectangles. The lengths of the sides are shown in centimetres. \includegraphics{figure_3}
  1. Find the coordinates of the centre of mass of the lamina, referred to the axes shown in Fig. 3.1. [5]
The rectangles BCJA and FGHI are folded through 90° about the lines CJ and FI respectively to give the fire-screen shown in Fig. 3.2.
  1. Show that the coordinates of the centre of mass of the fire-screen, referred to the axes shown in Fig. 3.2, are (2.5, 0, 57.5). [4]
The \(x\)- and \(y\)-axes are in a horizontal floor. The fire-screen has a weight of 72 N. A horizontal force \(P\) N is applied to the fire-screen at the point Z. This force is perpendicular to the line DE in the positive \(x\) direction. The fire-screen is on the point of tipping about the line AH.
  1. Calculate the value of \(P\). [5]
The coefficient of friction between the fire-screen and the floor is \(\mu\).
  1. For what values of \(\mu\) does the fire-screen slide before it tips? [4]
OCR MEI M2 2008 January Q4
18 marks Standard +0.3
Fig. 4.1 shows a uniform beam, CE, of weight 2200 N and length 4.5 m. The beam is freely pivoted on a fixed support at D and is supported at C. The distance CD is 2.75 m. \includegraphics{figure_4} The beam is horizontal and in equilibrium.
  1. Show that the anticlockwise moment of the weight of the beam about D is 1100 N m. Find the value of the normal reaction on the beam of the support at C. [6]
The support at C is removed and spheres at P and Q are suspended from the beam by light strings attached to the points C and R. The sphere at P has weight 440 N and the sphere at Q has weight \(W\) N. The point R of the beam is 1.5 m from D. This situation is shown in Fig. 4.2.
  1. The beam is horizontal and in equilibrium. Show that \(W = 1540\). [3]
The sphere at P is changed for a lighter one with weight 400 N. The sphere at Q is unchanged. The beam is now held in equilibrium at an angle of 20° to the horizontal by means of a light rope attached to the beam at E. This situation (but without the rope at E) is shown in Fig. 4.3. \includegraphics{figure_5}
  1. Calculate the tension in the rope when it is
    1. at 90° to the beam, [6]
    2. horizontal. [3]
OCR MEI M2 2011 January Q1
19 marks Standard +0.3
Fig. 1.1 shows block A of mass 2.5 kg which has been placed on a long, uniformly rough slope inclined at an angle \(\alpha\) to the horizontal, where \(\cos \alpha = 0.8\). The coefficient of friction between A and the slope is 0.85. \includegraphics{figure_1}
  1. Calculate the maximum possible frictional force between A and the slope. Show that A will remain at rest. [6]
With A still at rest, block B of mass 1.5 kg is projected down the slope, as shown in Fig. 1.2. B has a speed of 16 m s\(^{-1}\) when it collides with A. In this collision the coefficient of restitution is 0.4, the impulses are parallel to the slope and linear momentum parallel to the slope is conserved.
  1. Show that the velocity of A immediately after the collision is 8.4 m s\(^{-1}\) down the slope. Find the velocity of B immediately after the collision. [6]
  2. Calculate the impulse on B in the collision. [3]
The blocks do not collide again.
  1. For what length of time after the collision does A slide before it comes to rest? [4]
OCR MEI M2 2011 January Q2
17 marks Standard +0.3
  1. A firework is instantaneously at rest in the air when it explodes into two parts. One part is the body B of mass 0.06 kg and the other a cap C of mass 0.004 kg. The total kinetic energy given to B and C is 0.8 J. B moves off horizontally in the \(\mathbf{i}\) direction. By considering both kinetic energy and linear momentum, calculate the velocities of B and C immediately after the explosion. [8]
  2. A car of mass 800 kg is travelling up some hills. In one situation the car climbs a vertical height of 20 m while its speed decreases from 30 m s\(^{-1}\) to 12 m s\(^{-1}\). The car is subject to a resistance to its motion but there is no driving force and the brakes are not being applied.
    1. Using an energy method, calculate the work done by the car against the resistance to its motion. [4]
    In another situation the car is travelling at a constant speed of 18 m s\(^{-1}\) and climbs a vertical height of 20 m in 25 s up a uniform slope. The resistance to its motion is now 750 N.
    1. Calculate the power of the driving force required. [5]
OCR MEI M2 2011 January Q3
19 marks Standard +0.8
\includegraphics{figure_3} Fig. 3 shows a framework in equilibrium in a vertical plane. The framework is made from the equal, light, rigid rods AB, AD, BC, BD and CD so that ABD and BCD are equilateral triangles of side 2 m. AD and BC are horizontal. The rods are freely pin-jointed to each other at A, B, C and D. The pin-joint at A is fixed to a wall and the pin-joint at B rests on a smooth horizontal support. Fig. 3 also shows the external forces acting on the framework: there is a vertical load of 45 N at C and a horizontal force of 50 N applied at D; the normal reaction of the support on the framework at B is \(R\) N; horizontal and vertical forces \(X\) N and \(Y\) N act at A.
  1. Write down equations for the horizontal and vertical equilibrium of the framework. [2]
  2. Show that \(R = 135\) and \(Y = 90\). [3]
  3. On the diagram in your printed answer book, show the forces internal to the rods acting on the pin-joints. [2]
  4. Calculate the forces internal to the five rods, stating whether each rod is in tension or compression (thrust). [You may leave your answers in surd form. Your working in this part should correspond to your diagram in part (iii).] [10]
  5. Suppose that the force of magnitude 50 N applied at D is no longer horizontal, and the system remains in equilibrium in the same position. By considering the equilibrium at C, show that the forces in rods CD and BC are not changed. [2]
OCR MEI M2 2011 January Q4
17 marks Standard +0.3
You are given that the centre of mass, G, of a uniform lamina in the shape of an isosceles triangle lies on its axis of symmetry in the position shown in Fig. 4.1. \includegraphics{figure_4_1} Fig. 4.2 shows the cross-section OABCD of a prism made from uniform material. OAB is an isosceles triangle, where OA = AB, and OBCD is a rectangle. The distance OD is \(h\) cm, where \(h\) can take various positive values. All coordinates refer to the axes Ox and Oy shown. The units of the axes are centimetres. \includegraphics{figure_4_2}
  1. Write down the coordinates of the centre of mass of the triangle OAB. [1]
  2. Show that the centre of mass of the region OABCD is \(\left(\frac{12-h^2}{2(h+3)}, 2.5\right)\). [6]
The \(x\)-axis is horizontal. The prism is placed on a horizontal plane in the position shown in Fig. 4.2.
  1. Find the values of \(h\) for which the prism would topple. [3]
The following questions refer to the case where \(h = 3\) with the prism held in the position shown in Fig. 4.2. The cross-section OABCD contains the centre of mass of the prism. The weight of the prism is 15 N. You should assume that the prism does not slide.
  1. Suppose that the prism is held in this position by a vertical force applied at A. Given that the prism is on the point of tipping clockwise, calculate the magnitude of this force. [3]
  2. Suppose instead that the prism is held in this position by a force in the plane of the cross-section OABCD, applied at 30° below the horizontal at C, as shown in Fig. 4.3. Given that the prism is on the point of tipping anti-clockwise, calculate the magnitude of this force. [4]
\includegraphics{figure_4_3}
AQA M3 2016 June Q1
4 marks Moderate -0.8
At a firing range, a man holds a gun and fires a bullet horizontally. The bullet is fired with a horizontal velocity of \(400 \text{ m s}^{-1}\). The mass of the gun is \(1.5\) kg and the mass of the bullet is \(30\) grams.
  1. Find the speed of recoil of the gun. [2 marks]
  2. Find the magnitude of the impulse exerted by the man on the gun in bringing the gun to rest after the bullet is fired. [2 marks]
AQA M3 2016 June Q2
6 marks Moderate -0.8
A lunar mapping satellite of mass \(m_1\) measured in kg is in an elliptic orbit around the moon, which has mass \(m_2\) measured in kg. The effective potential, \(E\), of the satellite is given by $$E = \frac{K^2}{2m_1r^2} - \frac{Gm_1m_2}{r}$$ where \(r\) measured in metres is the distance of the satellite from the moon, \(G\) Nm\(^2\)kg\(^{-2}\) is the universal gravitational constant, and \(K\) is the angular momentum of the satellite. By using dimensional analysis, find the dimensions of:
  1. \(E\), [3 marks]
  2. \(K\). [3 marks]
AQA M3 2016 June Q3
12 marks Standard +0.8
A ball is projected from a point \(O\) on horizontal ground with speed \(14 \text{ m s}^{-1}\) at an angle of elevation \(30°\) above the horizontal. The ball travels in a vertical plane through the point \(O\) and hits a point \(Q\) on a plane which is inclined at \(45°\) to the horizontal. The point \(O\) is \(6\) metres from \(P\), the foot of the inclined plane, as shown in the diagram. The points \(O\), \(P\) and \(Q\) lie in the same vertical plane. The line \(PQ\) is a line of greatest slope of the inclined plane. \includegraphics{figure_3}
  1. During its flight, the horizontal and upward vertical distances of the ball from \(O\) are \(x\) metres and \(y\) metres respectively. Show that \(x\) and \(y\) satisfy the equation $$y = x\frac{\sqrt{3}}{3} - \frac{x^2}{30}$$ Use \(\cos 30° = \frac{\sqrt{3}}{2}\) and \(\tan 30° = \frac{\sqrt{3}}{3}\). [5 marks]
  2. Find the distance \(PQ\). [7 marks]
AQA M3 2016 June Q4
14 marks Standard +0.3
A smooth uniform sphere \(A\), of mass \(m\), is moving with velocity \(8u\) in a straight line on a smooth horizontal table. A smooth uniform sphere \(B\), of mass \(4m\), has the same radius as \(A\) and is moving on the table with velocity \(u\). \includegraphics{figure_4} The sphere \(A\) collides directly with the sphere \(B\). The coefficient of restitution between \(A\) and \(B\) is \(e\).
    1. Find, in terms of \(u\) and \(e\), the velocities of \(A\) and \(B\) immediately after the collision. [6 marks]
    2. The direction of motion of \(A\) is reversed by the collision. Show that \(e > a\), where \(a\) is a constant to be determined. [2 marks]
  2. Subsequently, \(B\) collides with a fixed smooth vertical wall which is at right angles to the direction of motion of \(A\) and \(B\). The coefficient of restitution between \(B\) and the wall is \(\frac{2}{5}\). The sphere \(B\) collides with \(A\) again after rebounding from the wall. Show that \(e < b\), where \(b\) is a constant to be determined. [3 marks]
  3. Given that \(e = \frac{4}{7}\), find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted on \(B\) by the wall. [3 marks]