Questions — OCR MEI M1 (268 questions)

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OCR MEI M1 Q5
5 A train consists of an engine of mass 10000 kg pulling one truck of mass 4000 kg . The coupling between the engine and the truck is light and parallel to the track. The train is accelerating at \(0.25 \mathrm {~m} \mathrm {~s} ^ { 2 }\) along a straight, level track.
  1. What is the resultant force on the train in the direction of its motion? The driving force of the engine is 4000 N .
  2. What is the resistance to the motion of the train?
  3. If the tension in the coupling is 1150 N , what is the resistance to the motion of the truck? With the same overall resistance to motion, the train now climbs a uniform slope inclined at \(3 ^ { \circ }\) to the horizontal with the same acceleration of \(0.25 \mathrm {~m} \mathrm {~s} ^ { 2 }\).
  4. What extra driving force is being applied?
OCR MEI M1 Q6
6 A box of weight 147 N is held by light strings AB and BC . As shown in Fig. 7.1, AB is inclined at \(\alpha\) to the horizontal and is fixed at \(\mathrm { A } ; \mathrm { BC }\) is held at C . The box is in equilibrium with BC horizontal and \(\alpha\) such that \(\sin \alpha = 0.6\) and \(\cos \alpha = 0.8\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bf477f61-9f8f-418a-86d8-392bc30323b1-4_380_542_377_791} \captionsetup{labelformat=empty} \caption{Fig. 7.1}
\end{figure}
  1. Calculate the tension in string AB .
  2. Show that the tension in string BC is 196 N . As shown in Fig. 7.2, a box of weight 90 N is now attached at C and another light string CD is held at D so that the system is in equilibrium with BC still horizontal. CD is inclined at \(\beta\) to the horizontal. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{bf477f61-9f8f-418a-86d8-392bc30323b1-4_378_695_1282_687} \captionsetup{labelformat=empty} \caption{Fig. 7.2}
    \end{figure}
  3. Explain why the tension in the string BC is still 196 N .
  4. Draw a diagram showing the forces acting on the box at C . Find the angle \(\beta\) and show that the tension in CD is 216 N , correct to three significant figures. The string section CD is now taken over a smooth pulley and attached to a block of mass \(M \mathrm {~kg}\) on a rough slope inclined at \(40 ^ { \circ }\) to the horizontal. As shown in Fig. 7.3, the part of the string attached to the box is still at \(\beta\) to the horizontal and the part attached to the block is parallel to the slope. The system is in equilibrium with a frictional force of 20 N acting on the block up the slope. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{bf477f61-9f8f-418a-86d8-392bc30323b1-5_436_1049_524_536} \captionsetup{labelformat=empty} \caption{Fig. 7.3}
    \end{figure}
  5. Calculate the value of \(M\).
OCR MEI M1 Q7
7 A block of mass 4 kg is in equilibrium on a rough plane inclined at \(60 ^ { \circ }\) to the horizontal, as shown in Fig. 4. A frictional force of 10 N acts up the plane and a vertical string AB attached to the block is in tension. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bf477f61-9f8f-418a-86d8-392bc30323b1-5_492_347_1545_870} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
  1. Draw a diagram showing the four forces acting on the block.
  2. By considering the components of the forces parallel to the slope, calculate the tension in the string.
  3. Calculate the normal reaction of the plane on the block.
OCR MEI M1 Q1
1 A golf ball is hit at an angle of \(60 ^ { \circ }\) to the horizontal from a point, O, on level horizontal ground. Its initial speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The standard projectile model, in which air resistance is neglected, is used to describe the subsequent motion of the golf ball. At time \(t \mathrm {~s}\) the horizontal and vertical components of its displacement from O are denoted by \(x \mathrm {~m}\) and \(y \mathrm {~m}\).
  1. Write down equations for \(x\) and \(y\) in terms of \(t\).
  2. Hence show that the equation of the trajectory is $$y = \sqrt { 3 } x - 0.049 x ^ { 2 }$$
  3. Find the range of the golf ball.
  4. A bird is hovering at position \(( 20,16 )\). Find whether the golf ball passes above it, passes below it or hits it.
OCR MEI M1 Q2
2 A football is kicked with speed \(31 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(20 ^ { \circ }\) to the horizontal. It travels towards the goal which is 50 m away. The height of the crossbar of the goal is 2.44 m .
  1. Does the ball go over the top of the crossbar? Justify your answer.
  2. State one assumption that you made in answering part (i).
OCR MEI M1 Q3
3 Fig. 1 shows the speed-time graph of a runner during part of his training. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bb65e726-a5e0-4060-81a6-6837dea82e64-2_1070_1588_319_273} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} For each of the following statements, say whether it is true or false. If it is false give a brief explanation.
(A) The graph shows that the runner finishes where he started.
(B) The runner's maximum speed is \(8 \mathrm {~ms} ^ { - 1 }\).
(C) At time 58 seconds, the runner is slowing down at a rate of \(1.6 \mathrm {~ms} ^ { - 2 }\).
(D) The runner travels 400 m altogether.
OCR MEI M1 Q4
4 A pellet is fired vertically upwards at a speed of \(11 \mathrm {~ms} ^ { - 1 }\). Assuming that air resistance may be neglected, calculate the speed at which the pellet hits a ceiling 2.4 m above its point of projection.
OCR MEI M1 Q5
5 Fig. 5 shows a block of mass 10 kg at rest on a rough horizontal floor. A light string, at an angle of \(30 ^ { \circ }\) to the vertical, is attached to the block. The tension in the string is 50 N . The block is in equilibrium. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bb65e726-a5e0-4060-81a6-6837dea82e64-3_397_577_567_795} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure}
  1. Show all the forces acting on the block.
  2. Show that the frictional force acting on the block is 25 N .
  3. Calculate the normal reaction of the floor on the block.
  4. Calculate the magnitude of the total force the floor is exerting on the block.
OCR MEI M1 Q6
6 A small ball is kicked off the edge of a jetty over a calm sea. Air resistance is negligible. Fig. 6 shows
  • the point of projection, O,
  • the initial horizontal and vertical components of velocity,
  • the point A on the jetty vertically below O and at sea level,
  • the height, OA , of the jetty above the sea.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bb65e726-a5e0-4060-81a6-6837dea82e64-4_451_1000_596_600} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure} The time elapsed after the ball is kicked is \(t\) seconds.
  1. Find an expression in terms of \(t\) for the height of the ball above O at time \(t\). Find also an expression for the horizontal distance of the ball from O at this time.
  2. Determine how far the ball lands from A .
OCR MEI M1 Q7
7 Fig. 4 shows a particle projected over horizontal ground from a point O at ground level. The particle initially has a speed of \(32 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) to the horizontal. The particle is a horizontal distance of 44.8 m from O after 5 seconds. Air resistance should be neglected. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bb65e726-a5e0-4060-81a6-6837dea82e64-5_562_757_389_729} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
  1. Write down an expression, in terms of \(\alpha\) and \(t\), for the horizontal distance of the particle from O at time \(t\) seconds after it is projected.
  2. Show that \(\cos \alpha = 0.28\).
  3. Calculate the greatest height reached by the particle.
OCR MEI M1 Q2
2 A ball is kicked from ground level over horizontal ground. It leaves the ground at a speed of 25 ms 1 and at an angle \(\theta\) to the horizontal such that \(\cos \theta = 0.96\) and \(\sin \theta = 0.28\).
  1. Show that the height, \(y \mathrm {~m}\), of the ball above the ground \(t\) seconds after projection is given by \(y = 7 t - 4.9 t ^ { 2 }\). Show also that the horizontal distance, \(x \mathrm {~m}\), travelled by this time is given by \(x = 24 t\).
  2. Calculate the maximum height reached by the ball.
  3. Calculate the times at which the ball is at half its maximum height. Find the horizontal distance travelled by the ball between these times.
  4. Determine the following when \(t = 1.25\).
    (A) The vertical component of the velocity of the ball.
    (B) Whether the ball is rising or falling. (You should give a reason for your answer.)
    (C) The speed of the ball.
  5. Show that the equation of the trajectory of the ball is $$y = \frac { 0.7 x } { 576 } ( 240 - 7 x )$$ Hence, or otherwise, find the range of the ball.
OCR MEI M1 Q3
3 A particle is thrown vertically upwards and returns to its point of projection after 6 seconds. Air resistance is negligible. Calculate the speed of projection of the particle and also the maximum height it reaches.
OCR MEI M1 Q4
4 You should neglect air resistance in this question.
A small stone is projected from ground level. The maximum height of the stone above horizontal ground is 22.5 m .
  1. Show that the vertical component of the initial velocity of the stone is \(21 \mathrm {~ms} { } ^ { 1 }\). The speed of projection is \(28 \mathrm {~ms} { } ^ { 1 }\).
  2. Find the angle of projection of the stone.
  3. Find the horizontal range of the stone.
OCR MEI M1 Q5
5 In this question take the value of \(\boldsymbol { g }\) to be \(\mathbf { 1 0 ~ } \mathbf { m ~ s } ^ { \mathbf { 2 } }\).
\(\Lambda\) particle \(\Lambda\) is projected over horizontal ground from a point P which is 9 m above a point O on the ground. The initial velocity has horizontal and vertical components of \(10 \mathrm {~ms} ^ { - 1 }\) and \(12 \mathrm {~ms} ^ { - 1 }\) respectively, as shown in Fig. 7. The trajectory of the particle meets the ground at X. Air resistance may be neglected. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9eab8ba4-d97b-4e3a-b36d-53f4bc7a80c2-3_394_788_551_630} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Calculate the specd of projection \(u \mathrm {~ms} ^ { - 1 }\) and the angle of projection \(\theta ^ { \circ }\).
  2. Show that, \(t\) seconds after projection, the height of particle A above the ground is \(9 + 12 t - 5 t ^ { 2 }\). Write down an expression in terms of \(t\) for the horizontal distance of the particle from O at this time.
  3. Calculate the maximum height of particle \(\Lambda\) above the point of projection.
  4. Calculate the distance OX .
    \(\wedge\) second particle, \(B\), is projected from \(O\) with speed \(20 \mathrm {~ms} ^ { - 1 }\) at \(60 ^ { \circ }\) to the horizontal. The trajectories of A and B are in the same vertical plane. Particles A and B are projected at the same time.
  5. Show that the horizontal displacements of A and B are always cqual.
  6. Show that, \(t\) seconds after projection, the height of particle B above the ground is \(10 \sqrt { 3 } t - 5 t ^ { 2 }\).
  7. Show that the particles collide 1.7 seconds after projection (correct to two significant figures).
OCR MEI M1 Q6
6 Ali is throwing flat stones onto water, hoping that they will bounce, as illustrated in Fig. 5.
Ali throws one stone from a height of 1.225 m above the water with initial speed \(20 \mathrm {~ms} ^ { - 1 }\) in a horizontal direction. Air resistance should be neglected. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9eab8ba4-d97b-4e3a-b36d-53f4bc7a80c2-4_233_959_482_575} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure}
  1. Find the time it takes for the stone to reach the water.
  2. Find the speed of the stone when it reaches the water and the angle its trajectory makes with the horizontal at this time.
OCR MEI M1 Q7
7 A projectile P travels in a vertical plane over level ground. Its position vector \(\mathbf { r }\) at time \(t\) seconds after projection is modelled by $$\mathbf { r } = \binom { x } { y } = \binom { 0 } { 5 } + \binom { 30 } { 40 } t - \binom { 0 } { 5 } t ^ { 2 }$$ where distances are in metres and the origin is a point on the level ground.
  1. Write down
    (A) the height from which P is projected,
    (B) the value of \(g\) in this model.
  2. Find the displacement of P from \(t = 3\) to \(t = 5\).
  3. Show that the equation of the trajectory is $$y = 5 + \frac { 4 } { 3 } x - \frac { x ^ { 2 } } { 180 }$$
OCR MEI M1 Q1
1 Fig. 7 shows the trajectory of an object which is projected from a point O on horizontal ground. Its initial velocity is \(40 \mathrm {~ms} ^ { - 1 }\) at an angle of \(\alpha\) to the horizontal. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7bcde451-5c86-4ed6-b6f5-62c1ad77618c-1_222_1246_267_439} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Show that, according to the standard projectile model in which air resistance is neglected, the flight time, \(T \mathrm {~s}\), and the range, \(R \mathrm {~m}\), are given by $$T = \frac { 80 \sin \alpha } { g } \text { and } R = \frac { 3200 \sin \alpha \cos \alpha } { g }$$ A company is designing a new type of ball and wants to model its flight.
  2. Initially the company uses the standard projectile model. Use this model to show that when \(\alpha = 30 ^ { \circ }\) and the initial speed is \(40 \mathrm {~ms} ^ { - 1 } , T\) is approximately 4.08 and \(R\) is approximately 141.4 . Find the values of \(T\) and \(R\) when \(\alpha = 45 ^ { \circ }\). The company tests the ball using a machine that projects it from ground level across horizontal ground. The speed of projection is set at \(40 \mathrm {~ms} ^ { - 1 }\). When the angle of projection is set at \(30 ^ { \circ }\), the range is found to be 125 m .
  3. Comment briefly on the accuracy of the standard projectile model in this situation. The company refines the model by assuming that the ball has a constant deceleration of \(2 \mathrm {~ms} ^ { - 2 }\) in the horizontal direction. In this new model, the resistance to the vertical motion is still neglected and so the flight time is still 4.08 s when the angle of projection is \(30 ^ { \circ }\).
  4. Using the new model, with \(\alpha = 30 ^ { \circ }\), show that the horizontal displacement from the point of projection, \(x \mathrm {~m}\) at time \(t \mathrm {~s}\), is given by $$x = 40 t \cos 30 ^ { \circ } - t ^ { 2 }$$ Find the range and hence show that this new model is reasonably accurate in this case. The company then sets the angle of projection to \(45 ^ { \circ }\) while retaining a projection speed of \(40 \mathrm {~ms} ^ { - 1 }\). With this setting the range of the ball is found to be 135 m .
  5. Investigate whether the new model is also accurate for this angle of projection.
  6. Make one suggestion as to how the model could be further refined. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{7bcde451-5c86-4ed6-b6f5-62c1ad77618c-2_722_1311_192_453} \captionsetup{labelformat=empty} \caption{Fig. 7}
    \end{figure} Fig. 7 shows a platform 10 m long and 2 m high standing on horizontal ground. A small ball projected from the surface of the platform at one end, O , just misses the other end, P . The ball is projected at \(68.5 ^ { \circ }\) to the horizontal with a speed of \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Air resistance may be neglected. At time \(t\) seconds after projection, the horizontal and vertical displacements of the ball from O are \(x \mathrm {~m}\) and \(y \mathrm {~m}\).
OCR MEI M1 Q5
5 Small stones A and B are initially in the positions shown in Fig. 6 with B a height \(H \mathrm {~m}\) directly above A. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7bcde451-5c86-4ed6-b6f5-62c1ad77618c-4_312_250_872_1004} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure} At the instant when B is released from rest, A is projected vertically upwards with a speed of \(29.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Air resistance may be neglected. The stones collide \(T\) seconds after they begin to move. At this instant they have the same speed, \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and A is still rising. By considering when the speed of A upwards is the same as the speed of B downwards, or otherwise, show that \(T = 1.5\) and find the values of \(V\) and \(H\).
OCR MEI M1 Q1
1 A small firework is fired from a point O at ground level over horizontal ground. The highest point reached by the firework is a horizontal distance of 60 m from O and a vertical distance of 40 m from O , as shown in Fig. 7. Air resistance is negligible. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{362d5995-bd39-4b07-b6a4-63eb1dd3e69d-1_611_1047_486_538} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure} The initial horizontal component of the velocity of the firework is \(21 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Calculate the time for the firework to reach its highest point and show that the initial vertical component of its velocity is \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Show that the firework is \(\left( 28 t - 4.9 t ^ { 2 } \right) \mathrm { m }\) above the ground \(t\) seconds after its projection. When the firework is at its highest point it explodes into several parts. Two of the parts initially continue to travel horizontally in the original direction, one with the original horizontal speed of \(21 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the other with a quarter of this speed.
  3. State why the two parts are always at the same height as one another above the ground and hence find an expression in terms of \(t\) for the distance between the parts \(t\) seconds after the explosion.
  4. Find the distance between these parts of the firework
    (A) when they reach the ground,
    (B) when they are 10 m above the ground.
  5. Show that the cartesian equation of the trajectory of the firework before it explodes is \(y = \frac { 1 } { 90 } \left( 120 x - x ^ { 2 } \right)\), referred to the coordinate axes shown in Fig. 7.
OCR MEI M1 Q1
1 A girl throws a small stone with initial speed \(14 \mathrm {~ms} { } ^ { 1 }\) at an angle of \(60 ^ { \circ }\) to the horizontal from a point 1 m above the ground. She throws the stone directly towards a vertical wall of height 6 m standing on horizontal ground. The point O is on the ground directly below the point of projection, as shown in Fig. 8. Air resistance is negligible. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4e0ddc86-c340-4057-bf3a-1c98587c3110-1_666_757_416_679} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Write down an expression in terms of \(t\) for the horizontal displacement of the stone from O , \(t\) seconds after projection. Find also an expression for the height of the stone above O at this time. The stone is at the top of its trajectory when it passes over the wall.
  2. (A) Find the time it takes for the stone to reach its highest point.
    (B) Calculate the distance of O from the base of the wall.
    (C) Show that the stone passes over the wall with 2.5 m clearance.
  3. Find the cartesian equation of the trajectory of the stone referred to the horizontal and vertical axes, \(\mathrm { O } x\) and \(\mathrm { O } y\). There is no need to simplify your answer. The girl now moves away a further distance \(d \mathrm {~m}\) from the wall. She throws a stone as before and it just passes over the wall.
  4. Calculate \(d\).
OCR MEI M1 Q2
2 A particle is projected vertically upwards from a point O at \(21 \mathrm {~ms} ^ { - 1 }\).
  1. Calculate the greatest height reached by the particle. When this particle is at its highest point, a second particle is projected vertically upwards from \(O\) at \(15 \mathrm {~ms} ^ { - 1 }\).
  2. Show that the particles collide 1.5 seconds later and determine the height above O at which the collision takes place.
OCR MEI M1 Q3
3 The trajectory ABCD of a small stone moving with negligible air resistance is shown in Fig. 7. AD is horizontal and BC is parallel to AD . The stone is projected from A with speed \(40 \mathrm {~ms} ^ { - 1 }\) at \(50 ^ { \circ }\) to the horizontal. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4e0ddc86-c340-4057-bf3a-1c98587c3110-3_316_1032_583_504} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Write down an expression for the horizontal displacement from A of the stone \(t\) seconds after projection. Write down also an expression for the vertical displacement at time \(t\).
  2. Show that the stone takes 6.253 seconds (to three decimal places) to travel from A to D . Calculate the range of the stone. You are given that \(X = 30\).
  3. Calculate the time it takes the stone to reach B . Hence determine the time for it to travel from A to C.
  4. Calculate the direction of the motion of the stone at C .
OCR MEI M1 Q1
1 The vectors \(\mathbf { P } , \mathbf { Q }\) and \(\mathbf { R }\) are given by $$\mathbf { P } = 5 \mathbf { i } + 4 \mathbf { j } , \quad \mathbf { Q } = 3 \mathbf { i } - 5 \mathbf { j } , \quad \mathbf { R } = - 8 \mathbf { i } + \mathbf { j }$$
  1. Find the vector \(\mathbf { P } + \mathbf { Q } + \mathbf { R }\).
  2. Interpret your answer to part (i) in the cases
    (A) \(\mathbf { P } , \mathbf { Q }\) and \(\mathbf { R }\) represent three forces acting on a particle,
    (B) \(\mathbf { P } , \mathbf { Q }\) and \(\mathbf { R }\) represent three stages of a hiker's walk.
OCR MEI M1 Q2
2 The vectors \(\mathbf { P } , \mathbf { Q }\) and \(\mathbf { R }\) are given by $$\mathbf { P } = 5 \mathbf { i } + 4 \mathbf { j } , \quad \mathbf { Q } = 3 \mathbf { i } - 5 \mathbf { j } , \quad \mathbf { R } = - 8 \mathbf { i } + \mathbf { j }$$
  1. Find the vector \(\mathbf { P } + \mathbf { Q } + \mathbf { R }\).
  2. Interpret your answer to part (i) in the cases
    (A) \(\mathbf { P } , \mathbf { Q }\) and \(\mathbf { R }\) represent three forces acting on a particle,
    (B) \(\mathbf { P } , \mathbf { Q }\) and \(\mathbf { R }\) represent three stages of a hiker's walk.
OCR MEI M1 Q3
3 In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are pointing east and north respectively.
  1. Calculate the bearing of the vector \(- 4 \mathbf { i } - 6 \mathbf { j }\). The vector \(- 4 \mathbf { i } - 6 \mathbf { j } + k ( 3 \mathbf { i } - 2 \mathbf { j } )\) is in the direction \(7 \mathbf { i } - 9 \mathbf { j }\).
  2. Find \(k\).