Questions M1 (1912 questions)

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OCR MEI M1 2015 June Q5
Moderate -0.3
5 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 \(20 \mathrm {~ms} ^ { - 1 }\). 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 2015 June Q6
Standard +0.3
6 The battery on Carol and Martin's car is flat so the car will not start. They hope to be able to "bump start" the car by letting it run down a hill and engaging the engine when the car is going fast enough. Fig. 6.1 shows the road leading away from their house, which is at A . The road is straight, and at all times the car is steered directly along it.
  • From A to B the road is horizontal.
  • Between B and C, it goes up a hill with a uniform slope of \(1.5 ^ { \circ }\) to the horizontal.
  • Between C and D the road goes down a hill with a uniform slope of \(3 ^ { \circ }\) to the horizontal. CD is 100 m . (This is the part of the road where they hope to get the car started.)
  • From D to E the road is again horizontal.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f87e062a-fdf2-45cf-8bc0-d05683b28e1a-4_241_1134_808_450} \captionsetup{labelformat=empty} \caption{Fig. 6.1}
\end{figure} The mass of the car is 750 kg , Carol's mass is 50 kg and Martin's mass is 80 kg .
Throughout the rest of this question, whenever Martin pushes the car, he exerts a force of 300 N along the line of the car.
  1. Between A and B , Martin pushes the car and Carol sits inside to steer it. The car has an acceleration of \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Show that the resistance to the car's motion is 100 N . Throughout the rest of this question you should assume that the resistance to motion is constant at 100 N .
  2. They stop at B and then Martin tries to push the car up the hill BC. Show that Martin cannot push the car up the hill with Carol inside it but can if she gets out.
    Find the acceleration of the car when Martin is pushing it and Carol is standing outside.
  3. While between B and C , Carol opens the window of the car and pushes it from outside while steering with one hand. Carol is able to exert a force of 150 N parallel to the surface of the road but at an angle of \(30 ^ { \circ }\) to the line of the car. This is illustrated in Fig. 6.2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f87e062a-fdf2-45cf-8bc0-d05683b28e1a-4_218_426_2133_831} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
    \end{figure} Find the acceleration of the car.
  4. At C, both Martin and Carol get in the car and, starting from rest, let it run down the hill under gravity. If the car reaches a speed of \(8 \mathrm {~ms} ^ { - 1 }\) they can get the engine to start. Does the car reach this speed before it reaches D ?
OCR MEI M1 2016 June Q1
Moderate -0.3
1 Fig. 1 shows a block of mass \(M \mathrm {~kg}\) being pushed over level ground by means of a light rod. The force, \(T \mathrm {~N}\), this exerts on the block is along the line of the rod. The ground is rough.
The rod makes an angle \(\alpha\) with the horizontal. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c8c96cf-5184-46e4-9c45-a8a80d0a6ff8-2_307_876_621_593} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure}
  1. Draw a diagram showing all the forces acting on the block.
  2. You are given that \(M = 5 , \alpha = 60 ^ { \circ } , T = 40\) and the acceleration of the block is \(1.5 \mathrm {~ms} ^ { - 2 }\). Find the frictional force.
OCR MEI M1 2016 June Q2
Moderate -0.3
2 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c8c96cf-5184-46e4-9c45-a8a80d0a6ff8-2_117_1162_1486_438} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} A particle moves on the straight line shown in Fig. 2. The positive direction is indicated on the diagram. The time, \(t\), is measured in seconds. The particle has constant acceleration, \(a \mathrm {~ms} ^ { - 2 }\). Initially it is at the point O and has velocity \(u \mathrm {~ms} ^ { - 1 }\).
When \(t = 2\), the particle is at A where OA is 12 m . The particle is also at A when \(t = 6\).
  1. Write down two equations in \(u\) and \(a\) and solve them.
  2. The particle changes direction when it is at B . Find the distance AB .
OCR MEI M1 2016 June Q3
Moderate -0.3
3 Fig. 3.1 shows a block of mass 8 kg on a smooth horizontal table.
This block is connected by a light string passing over a smooth pulley to a block of mass 4 kg which hangs freely. The part of the string between the 8 kg block and the pulley is parallel to the table. The system has acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c8c96cf-5184-46e4-9c45-a8a80d0a6ff8-3_330_809_525_628} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
\end{figure}
  1. Write down two equations of motion, one for each block.
  2. Find the value of \(a\). The table is now tilted at an angle of \(\theta\) to the horizontal as shown in Fig. 3.2. The system is set up as before; the 4 kg block still hangs freely. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{4c8c96cf-5184-46e4-9c45-a8a80d0a6ff8-3_410_727_1324_669} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
    \end{figure}
  3. The system is now in equilibrium. Find the value of \(\theta\).
OCR MEI M1 2016 June Q4
Moderate -0.8
4 A particle is initially at the origin, moving with velocity \(\mathbf { u }\). Its acceleration \(\mathbf { a }\) is constant. At time \(t\) its displacement from the origin is \(\mathbf { r } = \binom { x } { y }\), where \(\binom { x } { y } = \binom { 2 } { 6 } t - \binom { 0 } { 4 } t ^ { 2 }\).
  1. Write down \(\mathbf { u }\) and \(\mathbf { a }\) as column vectors.
  2. Find the speed of the particle when \(t = 2\).
  3. Show that the equation of the path of the particle is \(y = 3 x - x ^ { 2 }\).
OCR MEI M1 2016 June Q5
Standard +0.3
5 Mr McGregor is a keen vegetable gardener. A pigeon that eats his vegetables is his great enemy.
One day he sees the pigeon sitting on a small branch of a tree. He takes a stone from the ground and throws it. The trajectory of the stone is in a vertical plane that contains the pigeon. The same vertical plane intersects the window of his house. The situation is illustrated in Fig. 5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c8c96cf-5184-46e4-9c45-a8a80d0a6ff8-4_400_1221_1078_411} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure}
  • The stone is thrown from point O on level ground. Its initial velocity is \(15 \mathrm {~ms} ^ { - 1 }\) in the horizontal direction and \(8 \mathrm {~ms} ^ { - 1 }\) in the vertical direction.
  • The pigeon is at point P which is 4 m above the ground.
  • The house is 22.5 m from O .
  • The bottom of the window is 0.8 m above the ground and the window is 1.2 m high.
Show that the stone does not reach the height of the pigeon. Determine whether the stone hits the window.
OCR MEI M1 Q1
Moderate -0.8
1 Fig. 1 shows four forces acting at a point. The forces are in equilibrium. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{82f933a6-c17e-4b41-ae2b-3cc9d0ba975c-1_399_645_441_754} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Show that \(P = 14\).
Find \(Q\), giving your answer correct to 3 significant figures.
OCR MEI M1 Q2
Easy -1.2
2 Fig. 1 shows a pile of four uniform blocks in equilibrium on a horizontal table. Their masses, as shown, are \(4 \mathrm {~kg} , 5 \mathrm {~kg} , 7 \mathrm {~kg}\) and 10 kg . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{82f933a6-c17e-4b41-ae2b-3cc9d0ba975c-1_405_573_1560_777} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Mark on the diagram the magnitude and direction of each of the forces acting on the 7 kg block.
OCR MEI M1 Q3
Standard +0.3
3 Abi and Bob are standing on the ground and are trying to raise a small object of weight 250 N to the top of a building. They are using long light ropes. Fig. 7.1 shows the initial situation. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{82f933a6-c17e-4b41-ae2b-3cc9d0ba975c-2_770_1068_368_530} \captionsetup{labelformat=empty} \caption{Fig. 7.1}
\end{figure} Abi pulls vertically downwards on the rope A with a force \(F\) N. This rope passes over a small smooth pulley and is then connected to the object. Bob pulls on another rope, B, in order to keep the object away from the side of the building. In this situation, the object is stationary and in equilibrium. The tension in rope B, which is horizontal, is 25 N . The pulley is 30 m above the object. The part of rope A between the pulley and the object makes an angle \(\theta\) with the vertical.
  1. Represent the forces acting on the object as a fully labelled triangle of forces.
  2. Find \(F\) and \(\theta\). Show that the distance between the object and the vertical section of rope A is 3 m . Abi then pulls harder and the object moves upwards. Bob adjusts the tension in rope B so that the object moves along a vertical line. Fig. 7.2 shows the situation when the object is part of the way up. The tension in rope A is \(S \mathrm {~N}\) and the tension in rope B is \(T \mathrm {~N}\). The ropes make angles \(\alpha\) and \(\beta\) with the vertical as shown in the diagram. Abi and Bob are taking a rest and holding the object stationary and in equilibrium. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{82f933a6-c17e-4b41-ae2b-3cc9d0ba975c-3_384_357_520_851} \captionsetup{labelformat=empty} \caption{Fig. 7.2}
    \end{figure}
  3. Give the equations, involving \(S , T , \alpha\) and \(\beta\), for equilibrium in the vertical and horizontal directions.
  4. Find the values of \(S\) and \(T\) when \(\alpha = 8.5 ^ { \circ }\) and \(\beta = 35 ^ { \circ }\).
  5. Abi's mass is 40 kg . Explain why it is not possible for her to raise the object to a position in which \(\alpha = 60 ^ { \circ }\).
OCR MEI M1 Q4
Standard +0.3
4 Fig. 4 illustrates points \(A , B\) and \(C\) on a straight race track. The distance \(A B\) is 300 m and \(A C\) is 500 m .
A car is travelling along the track with uniform acceleration. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{82f933a6-c17e-4b41-ae2b-3cc9d0ba975c-4_70_1329_397_352} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} Initially the car is at A and travelling in the direction AB with speed \(5 \mathrm {~ms} ^ { - 1 }\). After 20 s it is at C .
  1. Find the acceleration of the car.
  2. Find the speed of the car at B and how long it takes to travel from A to B .
OCR MEI M1 Q5
Easy -1.2
5
An egg falls from rest a distance of 75 cm to the floor.
Neglecting air resistance, at what speed does it hit the floor?
OCR MEI M1 Q6
Moderate -0.8
6 Fig. 1 shows four forces in equilibrium. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{82f933a6-c17e-4b41-ae2b-3cc9d0ba975c-4_364_328_1748_901} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure}
  1. Find the value of \(P\).
  2. Hence find the value of \(Q\).
OCR MEI M1 Q7
Moderate -0.3
7 A block of weight 100 N is on a rough plane that is inclined at \(35 ^ { \circ }\) to the horizontal. The block is in equilibrium with a horizontal force of 40 N acting on it, as shown in Fig. 5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{82f933a6-c17e-4b41-ae2b-3cc9d0ba975c-5_490_880_316_623} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure} Calculate the frictional force acting on the block.
OCR MEI M1 Q1
Moderate -0.8
1 Fig. 2 shows two forces acting at A . The figure also shows the perpendicular unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) which are respectively horizontal and vertically upwards. The resultant of the two forces is \(\mathbf { F } \mathbf { N }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{94f23528-931c-47b6-89aa-4b6edd25cc30-1_264_918_584_663} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure}
  1. Find \(\mathbf { F }\) in terms of \(\mathbf { i }\) and \(\mathbf { j }\), giving your answer correct to three significant figures.
  2. Calculate the magnitude of \(\mathbf { F }\) and the angle that \(\mathbf { F }\) makes with the upward vertical.
OCR MEI M1 Q2
Moderate -0.8
2 Force \(\mathbf { F }\) is \(\left( \begin{array} { l } 4 \\ 1 \\ 2 \end{array} \right) \mathrm { N }\) and force \(\mathbf { G }\) is \(\left( \begin{array} { r } - 6 \\ 2 \\ 4 \end{array} \right) \mathrm { N }\).
  1. Find the resultant of \(\mathbf { F }\) and \(\mathbf { G }\) and calculate its magnitude.
  2. Forces \(\mathbf { F }\), \(2 \mathbf { G }\) and \(\mathbf { H }\) act on a particle which is in equilibrium. Find \(\mathbf { H }\).
OCR MEI M1 Q3
Moderate -0.8
3 A box of mass 5 kg is at rest on a rough horizontal floor.
  1. Find the value of the normal reaction of the floor on the box. The box remains at rest on the floor when a force of 10 N is applied to it at an angle of \(40 ^ { \circ }\) to the upward vertical, as shown in Fig. 3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{94f23528-931c-47b6-89aa-4b6edd25cc30-2_286_470_1067_803} \captionsetup{labelformat=empty} \caption{Fig. 3}
    \end{figure}
  2. Draw a diagram showing all the forces acting on the box.
  3. Calculate the new value of the normal reaction of the floor on the box and also the frictional force.
OCR MEI M1 Q4
Moderate -0.8
4 Fig. 4 shows forces of magnitudes 20 N and 16 N inclined at \(60 ^ { \circ }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{94f23528-931c-47b6-89aa-4b6edd25cc30-3_193_351_261_895} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
  1. Calculate the component of the resultant of these two forces in the direction of the 20 N force.
  2. Calculate the magnitude of the resultant of these two forces. These are the only forces acting on a particle of mass 2 kg .
  3. Find the magnitude of the acceleration of the particle and the angle the acceleration makes with the 20 N force.
OCR MEI M1 Q5
Moderate -0.8
5 A particle is in equilibrium when acted on by the forces \(\left( \begin{array} { r } x \\ - 7 \\ z \end{array} \right) , \left( \begin{array} { r } 4 \\ y \\ - 5 \end{array} \right)\) and \(\left( \begin{array} { r } 5 \\ 4 \\ - 7 \end{array} \right)\), where the units are newtons.
  1. Find the values of \(x , y\) and \(z\).
  2. Calculate the magnitude of \(\left( \begin{array} { r } 5 \\ 4 \\ - 7 \end{array} \right)\).
OCR MEI M1 Q6
Moderate -0.8
6 A small box B of weight 400 N is held in equilibrium by two light strings AB and BC . The string \(B C\) is fixed at \(C\). The end \(A\) of string \(A B\) is fixed so that \(A B\) is at an angle \(\alpha\) to the vertical where \(\alpha < 60 ^ { \circ }\). String BC is at \(60 ^ { \circ }\) to the vertical. This information is shown in Fig. 5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{94f23528-931c-47b6-89aa-4b6edd25cc30-4_404_437_434_810} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure}
  1. Draw a labelled diagram showing all the forces acting on the box.
  2. In one situation string AB is fixed so that \(\alpha = 30 ^ { \circ }\). By drawing a triangle of forces, or otherwise, calculate the tension in the string BC and the tension in the string AB .
  3. Show carefully, but briefly, that the box cannot be in equilibrium if \(\alpha = 60 ^ { \circ }\) and BC remains at \(60 ^ { \circ }\) to the vertical.
Edexcel M1 2013 January Q1
Moderate -0.3
  1. Two particles \(P\) and \(Q\) have masses \(4 m\) and \(m\) respectively. The particles are moving towards each other on a smooth horizontal plane and collide directly. The speeds of \(P\) and \(Q\) immediately before the collision are \(2 u\) and \(5 u\) respectively. Immediately after the collision, the speed of \(P\) is \(\frac { 1 } { 2 } u\) and its direction of motion is reversed.
    1. Find the speed and direction of motion of \(Q\) after the collision.
    2. Find the magnitude of the impulse exerted on \(P\) by \(Q\) in the collision.
    3. A steel girder \(A B\), of mass 200 kg and length 12 m , rests horizontally in equilibrium on two smooth supports at \(C\) and at \(D\), where \(A C = 2 \mathrm {~m}\) and \(D B = 2 \mathrm {~m}\). A man of mass 80 kg stands on the girder at the point \(P\), where \(A P = 4 \mathrm {~m}\), as shown in Figure 1.
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6e887c0f-911a-4f18-81a1-2f90025f5410-03_318_1061_420_440} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} The man is modelled as a particle and the girder is modelled as a uniform rod.
  2. Find the magnitude of the reaction on the girder at the support at \(C\). The support at \(D\) is now moved to the point \(X\) on the girder, where \(X B = x\) metres. The man remains on the girder at \(P\), as shown in Figure 2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6e887c0f-911a-4f18-81a1-2f90025f5410-03_324_1058_1292_443} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Given that the magnitudes of the reactions at the two supports are now equal and that the girder again rests horizontally in equilibrium, find
  3. the magnitude of the reaction at the support at \(X\),
  4. the value of \(x\).
Edexcel M1 2013 January Q3
Moderate -0.8
3. A particle \(P\) of mass 2 kg is attached to one end of a light string, the other end of which is attached to a fixed point \(O\). The particle is held in equilibrium, with \(O P\) at \(30 ^ { \circ }\) to the downward vertical, by a force of magnitude \(F\) newtons. The force acts in the same vertical plane as the string and acts at an angle of \(30 ^ { \circ }\) to the horizontal, as shown in Figure 3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6e887c0f-911a-4f18-81a1-2f90025f5410-05_444_510_477_699} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Find
  1. the value of \(F\),
  2. the tension in the string.
Edexcel M1 2013 January Q4
Standard +0.3
4. A lifeboat slides down a straight ramp inclined at an angle of \(15 ^ { \circ }\) to the horizontal. The lifeboat has mass 800 kg and the length of the ramp is 50 m . The lifeboat is released from rest at the top of the ramp and is moving with a speed of \(12.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it reaches the end of the ramp. By modelling the lifeboat as a particle and the ramp as a rough inclined plane, find the coefficient of friction between the lifeboat and the ramp.
(9)
Edexcel M1 2013 January Q5
Standard +0.3
5．\(v \left( \mathrm {~m} \mathrm {~s} ^ { - 1 } \right) ^ { \wedge }\) Figure 4 The velocity－time graph in Figure 4 represents the journey of a train \(P\) travelling along a straight horizontal track between two stations which are 1.5 km apart．The train \(P\) leaves the first station，accelerating uniformly from rest for 300 m until it reaches a speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) ．The train then maintains this speed for \(T\) seconds before decelerating uniformly at \(1.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) ，coming to rest at the next station．
（a）Find the acceleration of \(P\) during the first 300 m of its journey．
（b）Find the value of \(T\) ． A second train \(Q\) completes the same journey in the same total time．The train leaves the first station，accelerating uniformly from rest until it reaches a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and then immediately decelerates uniformly until it comes to rest at the next station．
（c）Sketch on the diagram above，a velocity－time graph which represents the journey of train \(Q\) ．
（d）Find the value of \(V\) ．
Edexcel M1 2013 January Q6
Moderate -0.3
6. [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively and position vectors are given with respect to a fixed origin.] A ship sets sail at 9 am from a port \(P\) and moves with constant velocity. The position vector of \(P\) is \(( 4 \mathbf { i } - 8 \mathbf { j } ) \mathrm { km }\). At 9.30 am the ship is at the point with position vector \(( \mathbf { i } - 4 \mathbf { j } ) \mathrm { km }\).
  1. Find the speed of the ship in \(\mathrm { km } \mathrm { h } ^ { - 1 }\).
  2. Show that the position vector \(\mathbf { r } \mathrm { km }\) of the ship, \(t\) hours after 9 am , is given by \(\mathbf { r } = ( 4 - 6 t ) \mathbf { i } + ( 8 t - 8 ) \mathbf { j }\). At 10 am , a passenger on the ship observes that a lighthouse \(L\) is due west of the ship. At 10.30 am , the passenger observes that \(L\) is now south-west of the ship.
  3. Find the position vector of \(L\).