Questions — OCR MEI M1 (276 questions)

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OCR MEI M1 Q4
7 marks Moderate -0.3
4 As shown in Fig. 4, boxes P and Q are descending vertically supported by a parachute. Box P has mass 75 kg . Box Q has mass 25 kg and hangs from box P by means of a light vertical wire. Air resistance on the boxes should be neglected. At one stage the boxes are slowing in their descent with the parachute exerting an upward vertical force of 1030 N on box P . The acceleration of the boxes is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) upwards and the tension in the wire is \(T \mathrm {~N}\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5a1895e1-abe3-4739-876a-f19458f0f6ed-3_332_358_1504_1526} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
  1. Draw a labelled diagram showing all the forces acting on box P and another diagram showing all the forces acting on box Q .
  2. Write down separate equations of motion for box P and for box Q .
  3. Calculate the tension in the wire.
OCR MEI M1 Q5
20 marks Standard +0.3
5 A cylindrical tub of mass 250 kg is on a horizontal floor. Resistance to its motion other than that due to friction is negligible. The first attempt to move the tub is by pulling it with a force of 150 N in the \(\mathbf { i }\) direction, as shown in Fig. 8.1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5a1895e1-abe3-4739-876a-f19458f0f6ed-4_310_1349_451_435} \captionsetup{labelformat=empty} \caption{Fig. 5.1}
\end{figure}
  1. Calculate the acceleration of the tub if friction is ignored. In fact, there is friction and the tub does not move.
  2. Write down the magnitude and direction of the frictional force opposing the pull. Two more forces are now added to the 150 N force in a second attempt to move the tub, as shown in Fig. 8.2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5a1895e1-abe3-4739-876a-f19458f0f6ed-4_497_927_1350_646} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
    \end{figure} Angle \(\theta\) is acute and chosen so that the resultant of the three forces is in the \(\mathbf { i }\) direction.
  3. Determine the value of \(\theta\) and the resultant of the three forces. With this resultant force, the tub moves with constant acceleration and travels 1 metre from rest in 2 seconds.
  4. Show that the magnitude of the friction acting on the tub is 661 N , correct to 3 significant figures. When the speed of the tub is \(1.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it comes to a part of the floor where the friction on the tub is 200 N greater. The pulling forces stay the same.
  5. Find the velocity of the tub when it has moved a further 1.65 m .
OCR MEI M1 Q6
16 marks Standard +0.3
6 An empty open box of mass 4 kg is on a plane that is inclined at \(25 ^ { \circ }\) to the horizontal.
In one model the plane is taken to be smooth. The box is held in equilibrium by a string with tension \(T \mathrm {~N}\) parallel to the plane, as shown in Fig. 6.1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5a1895e1-abe3-4739-876a-f19458f0f6ed-5_308_561_559_828} \captionsetup{labelformat=empty} \caption{Fig. 6.1}
\end{figure}
  1. Calculate \(T\). A rock of mass \(m \mathrm {~kg}\) is now put in the box. The system is in equilibrium when the tension in the string, still parallel to the plane, is 50 N .
  2. Find \(m\). In a refined model the plane is rough. The empty box, of mass 4 kg , is in equilibrium when a frictional force of 20 N acts down the plane and the string has a tension of \(P \mathrm {~N}\) inclined at \(15 ^ { \circ }\) to the plane, as shown in Fig. 6.2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5a1895e1-abe3-4739-876a-f19458f0f6ed-5_359_559_1599_830} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
    \end{figure}
  3. Draw a diagram showing all the forces acting on the box.
  4. Calculate \(P\).
  5. Calculate the normal reaction of the plane on the box.
OCR MEI M1 Q1
8 marks Standard +0.3
1 Fig. 1.1 shows a circular cylinder of mass 100 kg being raised by a light, inextensible vertical wire AB . There is negligible air resistance. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bf477f61-9f8f-418a-86d8-392bc30323b1-1_306_256_368_965} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
\end{figure}
  1. Calculate the acceleration of the cylinder when the tension in the wire is 1000 N .
  2. Calculate the tension in the wire when the cylinder has an upward acceleration of \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The cylinder is now raised inside a fixed smooth vertical tube that prevents horizontal motion but provides negligible resistance to the upward motion of the cylinder. When the wire is inclined at \(30 ^ { \circ }\) to the vertical, as shown in Fig. 1.2, the cylinder again has an upward acceleration of \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{bf477f61-9f8f-418a-86d8-392bc30323b1-1_308_490_1230_849} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
    \end{figure}
  3. Calculate the new tension in the wire.
OCR MEI M1 Q2
4 marks Moderate -0.8
2 Boxes A and B slide on a smooth, horizontal plane. Box A has a mass of 4 kg and box B a mass of 5 kg . They are connected by a light, inextensible, horizontal wire. Horizontal forces of 9 N and 135 N act on A and B in the directions shown in Fig. 5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bf477f61-9f8f-418a-86d8-392bc30323b1-1_95_915_2042_650} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure} Calculate the tension in the wire joining the boxes.
OCR MEI M1 Q3
4 marks Standard +0.3
3 Fig. 3 shows a system in equilibrium. The rod is firmly attached to the floor and also to an object, P. The light string is attached to P and passes over a smooth pulley with an object Q hanging freely from its other end. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bf477f61-9f8f-418a-86d8-392bc30323b1-2_519_629_370_745} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
  1. Why is the tension the same throughout the string?
  2. Calculate the force in the rod, stating whether it is a tension or a thrust.
OCR MEI M1 Q4
7 marks Moderate -0.3
4 Two trucks, A and B, each of mass 10000 kg , are pulled along a straight, horizontal track by a constant, horizontal force of \(P \mathrm {~N}\). The coupling between the trucks is light and horizontal. This situation and the resistances to motion of the trucks are shown in Fig. 4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bf477f61-9f8f-418a-86d8-392bc30323b1-2_198_948_1454_592} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} The acceleration of the system is \(0.2 \mathrm {~ms} ^ { 2 }\) in the direction of the pulling force of magnitude \(P\).
  1. Calculate the value of \(P\). Truck A is now subjected to an extra resistive force of 2000 N while \(P\) does not change.
  2. Calculate the new acceleration of the trucks.
  3. Calculate the force in the coupling between the trucks.
OCR MEI M1 Q5
8 marks Moderate -0.8
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
14 marks Standard +0.3
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 marks Moderate -0.3
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
8 marks Moderate -0.3
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
7 marks Moderate -0.3
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
6 marks Moderate -0.8
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
3 marks Easy -1.2
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
8 marks Moderate -0.3
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
8 marks Moderate -0.8
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 marks Standard +0.3
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
19 marks Moderate -0.3
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
4 marks Moderate -0.8
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
8 marks Moderate -0.8
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
19 marks Standard +0.3
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
7 marks Moderate -0.8
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
8 marks Moderate -0.8
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
19 marks Standard +0.3
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
20 marks Standard +0.3
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 \(d\).