Questions M1 (1912 questions)

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OCR MEI M1 2012 January Q6
18 marks Standard +0.3
6 Robin is driving a car of mass 800 kg along a straight horizontal road at a speed of \(40 \mathrm {~ms} ^ { - 1 }\).
Robin applies the brakes and the car decelerates uniformly; it comes to rest after travelling a distance of 125 m .
  1. Show that the resistance force on the car when the brakes are applied is 5120 N .
  2. Find the time the car takes to come to rest. For the rest of this question, assume that when Robin applies the brakes there is a constant resistance force of 5120 N on the car. The car returns to its speed of \(40 \mathrm {~ms} ^ { - 1 }\) and the road remains straight and horizontal.
    Robin sees a red light 155 m ahead, takes a short time to react and then applies the brakes.
    The car comes to rest before it reaches the red light.
  3. Show that Robin's reaction time is less than 0.75 s . The 'stopping distance' is the total distance travelled while a driver reacts and then applies the brakes to bring the car to rest. For the rest of this question, assume that Robin is still driving the car described above and has a reaction time of 0.675 s . (This is the figure used in calculating the stopping distances given in the Highway Code.)
  4. Calculate the stopping distance when Robin is driving at \(20 \mathrm {~ms} ^ { - 1 }\) on a horizontal road. The car then travels down a hill which has a slope of \(5 ^ { \circ }\) to the horizontal.
  5. Find the stopping distance when Robin is driving at \(20 \mathrm {~ms} ^ { - 1 }\) down this hill.
  6. By what percentage is the stopping distance increased by the fact that the car is going down the hill? Give your answer to the nearest \(1 \%\).
OCR MEI M1 2013 January Q1
6 marks Moderate -0.8
1 Fig. 1 shows a block of mass 3 kg on a plane which is inclined at an angle of \(30 ^ { \circ }\) to the horizontal.
A force \(P \mathrm {~N}\) is applied to the block parallel to the plane in the upwards direction.
The plane is rough so that a frictional force of 10 N opposes the motion.
The block is moving at constant speed up the plane. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{13f555cc-d506-48e5-a0e4-225cae4251dc-3_214_622_657_724} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure}
  1. Mark and label all the forces acting on the block.
  2. Calculate the magnitude of the normal reaction of the plane on the block.
  3. Calculate the magnitude of the force \(P\).
OCR MEI M1 2013 January Q2
8 marks Moderate -0.8
2 In this question, the unit vectors \(\binom { 1 } { 0 }\) and \(\binom { 0 } { 1 }\) are in the directions east and north.
Distance is measured in metres and time, \(t\), in seconds.
A radio-controlled toy car moves on a flat horizontal surface. A child is standing at the origin and controlling the car.
When \(t = 0\), the displacement of the car from the origin is \(\binom { 0 } { - 2 } \mathrm {~m}\), and the car has velocity \(\binom { 2 } { 0 } \mathrm {~ms} ^ { - 1 }\). The acceleration of the car is constant and is \(\binom { - 1 } { 1 } \mathrm {~ms} ^ { - 2 }\).
  1. Find the velocity of the car at time \(t\) and its speed when \(t = 8\).
  2. Find the distance of the car from the child when \(t = 8\).
OCR MEI M1 2013 January Q3
8 marks Moderate -0.8
3 Fig. 3 shows two people, Sam and Tom, pushing a car of mass 1000 kg along a straight line \(l\) on level ground. Sam pushes with a constant horizontal force of 300 N at an angle of \(30 ^ { \circ }\) to the line \(l\).
Tom pushes with a constant horizontal force of 175 N at an angle of \(15 ^ { \circ }\) to the line \(l\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{13f555cc-d506-48e5-a0e4-225cae4251dc-4_291_1132_534_479} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
  1. The car starts at rest and moves with constant acceleration. After 6 seconds it has travelled 7.2 m . Find its acceleration.
  2. Find the resistance force acting on the car along the line \(l\).
  3. The resultant of the forces exerted by Sam and Tom is not in the direction of the car's acceleration. Explain briefly why.
OCR MEI M1 2013 January Q4
7 marks Standard +0.3
4 A particle is travelling along a straight line with constant acceleration. \(\mathrm { P } , \mathrm { O }\) and Q are points on the line, as illustrated in Fig. 4. The distance from P to O is 5 m and the distance from O to Q is 30 m . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{13f555cc-d506-48e5-a0e4-225cae4251dc-4_113_1173_1576_447} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} Initially the particle is at O . After 10 s , it is at Q and its velocity is \(9 \mathrm {~ms} ^ { - 1 }\) in the direction \(\overrightarrow { \mathrm { OQ } }\).
  1. Find the initial velocity and the acceleration of the particle.
  2. Prove that the particle is never at P .
OCR MEI M1 2013 January Q5
7 marks Moderate -0.8
5 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]{13f555cc-d506-48e5-a0e4-225cae4251dc-5_229_953_434_557} \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 2013 January Q6
18 marks Moderate -0.3
6 The speed of a 100 metre runner in \(\mathrm { ms } ^ { - 1 }\) is measured electronically every 4 seconds.
The measurements are plotted as points on the speed-time graph in Fig. 6. The vertical dotted line is drawn through the runner's finishing time. Fig. 6 also illustrates Model P in which the points are joined by straight lines. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{13f555cc-d506-48e5-a0e4-225cae4251dc-6_1025_1504_641_260} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure}
  1. Use Model P to estimate
    (A) the distance the runner has gone at the end of 12 seconds,
    (B) how long the runner took to complete 100 m . A mathematician proposes Model Q in which the runner's speed, \(v \mathrm {~ms} ^ { - 1 }\) at time \(t \mathrm {~s}\), is given by $$v = \frac { 5 } { 2 } t - \frac { 1 } { 8 } t ^ { 2 }$$
  2. Verify that Model Q gives the correct speed for \(t = 8\).
  3. Use Model Q to estimate the distance the runner has gone at the end of 12 seconds.
  4. The runner was timed at 11.35 seconds for the 100 m . Which model places the runner closer to the finishing line at this time?
  5. Find the greatest acceleration of the runner according to each model.
OCR MEI M1 2005 June Q1
8 marks Moderate -0.8
1 A particle travels along a straight line. Its acceleration during the time interval \(0 \leqslant t \leqslant 8\) is given by the acceleration-time graph in Fig. 1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{04848aba-9e64-4265-a4a5-e9336b958a05-2_737_1274_502_461} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure}
  1. Write down the acceleration of the particle when \(t = 4\). Given that the particle starts from rest, find its speed when \(t = 4\).
  2. Write down an expression in terms of \(t\) for the acceleration, \(a \mathrm {~ms} ^ { - 2 }\), of the particle in the time interval \(0 \leqslant t \leqslant 4\).
  3. Without calculation, state the time at which the speed of the particle is greatest. Give a reason for your answer.
  4. Calculate the change in speed of the particle from \(t = 5\) to \(t = 8\), indicating whether this is an increase or a decrease.
OCR MEI M1 2005 June Q2
8 marks Moderate -0.8
2 A particle moves along the \(x\)-axis with velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at time \(t\) given by $$v = 24 t - 6 t ^ { 2 }$$ The positive direction is in the sense of \(x\) increasing.
  1. Find an expression for the acceleration of the particle at time \(t\).
  2. Find the times, \(t _ { 1 }\) and \(t _ { 2 }\), at which the particle has zero speed.
  3. Find the distance travelled between the times \(t _ { 1 }\) and \(t _ { 2 }\).
OCR MEI M1 2005 June Q3
6 marks Moderate -0.8
3 A particle rests on a smooth, horizontal plane. Horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) lie in this plane. The particle is in equilibrium under the action of the three forces \(( - 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { N }\) and \(( 21 \mathbf { i } - 7 \mathbf { j } ) \mathrm { N }\) and \(\mathbf { R N }\).
  1. Write down an expression for \(\mathbf { R }\) in terms of \(\mathbf { i }\) and \(\mathbf { j }\).
  2. Find the magnitude of \(\mathbf { R }\) and the angle between \(\mathbf { R }\) and the \(\mathbf { i }\) direction.
OCR MEI M1 2005 June Q4
7 marks Moderate -0.3
4 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]{04848aba-9e64-4265-a4a5-e9336b958a05-3_533_378_852_831} \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 2005 June Q5
7 marks Moderate -0.3
5 The position vector of a particle at time \(t\) is given by $$\mathbf { r } = \frac { 1 } { 2 } t \mathbf { i } + \left( t ^ { 2 } - 1 \right) \mathbf { j } ,$$ referred to an origin \(\mathbf { O }\) where \(\mathbf { i }\) and \(\mathbf { j }\) are the standard unit vectors in the directions of the cartesian axes \(\mathrm { O } x\) and Oy respectively.
  1. Write down the value of \(t\) for which the \(x\)-coordinate of the position of the particle is 2 . Find the \(y\)-coordinate at this time.
  2. Show that the cartesian equation of the path of the particle is \(y = 4 x ^ { 2 } - 1\).
  3. Find the coordinates of the point where the particle is moving at \(45 ^ { \circ }\) to both \(\mathrm { O } x\) and \(\mathrm { O } y\). Section B (36 marks)
OCR MEI M1 2006 June Q1
4 marks Moderate -0.8
1 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 2006 June Q2
8 marks Moderate -0.8
2 Force \(\mathbf { F } _ { 1 }\) is \(\binom { - 6 } { 13 } \mathrm {~N}\) and force \(\mathbf { F } _ { 2 }\) is \(\binom { - 3 } { 5 } \mathrm {~N}\), where \(\binom { 1 } { 0 }\) and \(\binom { 0 } { 1 }\) are vectors east and north respectively.
  1. Calculate the magnitude of \(\mathbf { F } _ { 1 }\), correct to three significant figures.
  2. Calculate the direction of the force \(\mathbf { F } _ { 1 } - \mathbf { F } _ { 2 }\) as a bearing. Force \(\mathbf { F } _ { 2 }\) is the resultant of all the forces acting on an object of mass 5 kg .
  3. Calculate the acceleration of the object and the change in its velocity after 10 seconds.
OCR MEI M1 2006 June Q3
10 marks Moderate -0.8
3 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 2006 June Q4
8 marks Moderate -0.8
4 Fig. 4 shows the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) in the directions of the cartesian axes \(\mathrm { O } x\) and \(\mathrm { O } y\), respectively. O is the origin of the axes and of position vectors. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4957086c-fd1c-4cdc-bbdb-1959b3b21b2d-3_383_383_424_840} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} The position vector of a particle is given by \(\mathbf { r } = 3 t \mathbf { i } + \left( 18 t ^ { 2 } - 1 \right) \mathbf { j }\) for \(t \geqslant 0\), where \(t\) is time.
  1. Show that the path of the particle cuts the \(x\)-axis just once.
  2. Find an expression for the velocity of the particle at time \(t\). Deduce that the particle never travels in the j direction.
  3. Find the cartesian equation of the path of the particle, simplifying your answer.
OCR MEI M1 2006 June Q5
8 marks Moderate -0.8
5 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 \(21 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The speed of projection is \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the angle of projection of the stone.
  3. Find the horizontal range of the stone. Section B (36 marks)
OCR MEI M1 2006 June Q6
18 marks Moderate -0.3
6 A toy car is travelling in a straight horizontal line.
One model of the motion for \(0 \leqslant t \leqslant 8\), where \(t\) is the time in seconds, is shown in the velocity-time graph Fig. 6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4957086c-fd1c-4cdc-bbdb-1959b3b21b2d-4_474_1196_580_424} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure}
  1. Calculate the distance travelled by the car from \(t = 0\) to \(t = 8\).
  2. How much less time would the car have taken to travel this distance if it had maintained its initial speed throughout?
  3. What is the acceleration of the car when \(t = 1\) ? From \(t = 8\) to \(t = 14\), the car travels 58.5 m with a new constant acceleration, \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  4. Find \(a\). A second model for the velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of the toy car is $$v = 12 - 10 t + \frac { 9 } { 4 } t ^ { 2 } - \frac { 1 } { 8 } t ^ { 3 } , \text { for } 0 \leqslant t \leqslant 8$$ This model agrees with the values for \(v\) given in Fig. 6 for \(t = 0,2,4\) and 6. [Note that you are not required to verify this.] Use this second model to answer the following questions.
  5. Calculate the acceleration of the car when \(t = 1\).
  6. Initially the car is at A. Find an expression in terms of \(t\) for the displacement of the car from A after the first \(t\) seconds of its motion. Hence find the displacement of the car from A when \(t = 8\).
  7. Explain with a reason what this model predicts for the motion of the car between \(t = 2\) and \(t = 4\).
OCR MEI M1 2006 June Q7
18 marks Standard +0.3
7 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 A ; 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]{4957086c-fd1c-4cdc-bbdb-1959b3b21b2d-5_381_547_440_753} \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]{4957086c-fd1c-4cdc-bbdb-1959b3b21b2d-5_387_702_1343_646} \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]{4957086c-fd1c-4cdc-bbdb-1959b3b21b2d-6_430_1045_493_502} \captionsetup{labelformat=empty} \caption{Fig. 7.3}
    \end{figure}
  5. Calculate the value of \(M\).
OCR MEI M1 2007 June Q1
5 marks Moderate -0.8
1 Fig. 1 shows four forces in equilibrium. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3be85526-3872-42ac-8278-1d4a3cf75ff7-2_369_332_413_868} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure}
  1. Find the value of \(P\).
  2. Hence find the value of \(Q\).
OCR MEI M1 2007 June Q2
8 marks Moderate -0.8
2 A car passes a point A travelling at \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Its motion over the next 45 seconds is modelled as follows.
  • The car's speed increases uniformly from \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over the first 10 s .
  • Its speed then increases uniformly to \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over the next 15 s .
  • The car then maintains this speed for a further 20 s at which time it reaches the point B .
    1. Sketch a speed-time graph to represent this motion.
    2. Calculate the distance from A to B .
    3. When it reaches the point B , the car is brought uniformly to rest in \(T\) seconds. The total distance from A is now 1700 m . Calculate the value of \(T\).
OCR MEI M1 2007 June Q3
4 marks Moderate -0.8
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]{3be85526-3872-42ac-8278-1d4a3cf75ff7-3_526_633_429_708} \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 2007 June 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]{3be85526-3872-42ac-8278-1d4a3cf75ff7-3_205_958_1516_552} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} The acceleration of the system is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) 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 2007 June Q5
4 marks Moderate -0.3
5 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]{3be85526-3872-42ac-8278-1d4a3cf75ff7-4_490_874_379_591} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure} Calculate the frictional force acting on the block.
OCR MEI M1 2007 June Q6
8 marks Moderate -0.3
6 A rock of mass 8 kg is acted on by just the two forces \(- 80 \mathbf { k } \mathrm {~N}\) and \(( - \mathbf { i } + 16 \mathbf { j } + 72 \mathbf { k } ) \mathrm { N }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane and \(\mathbf { k }\) is a unit vector vertically upward.
  1. Show that the acceleration of the rock is \(\left( - \frac { 1 } { 8 } \mathbf { i } + 2 \mathbf { j } - \mathbf { k } \right) \mathrm { ms } ^ { - 2 }\). The rock passes through the origin of position vectors, O , with velocity \(( \mathbf { i } - 4 \mathbf { j } + 3 \mathbf { k } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and 4 seconds later passes through the point A .
  2. Find the position vector of A .
  3. Find the distance OA .
  4. Find the angle that OA makes with the horizontal. Section B (36 marks)