Questions — OCR MEI M1 (268 questions)

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OCR MEI M1 2005 June Q3
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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)
OCR MEI M1 2007 June Q7
7 Fig. 7 is a sketch of part of the velocity-time graph for the motion of an insect walking in a straight line. Its velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at time \(t\) seconds for the time interval \(- 3 \leqslant t \leqslant 5\) is given by $$v = t ^ { 2 } - 2 t - 8 .$$ \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3be85526-3872-42ac-8278-1d4a3cf75ff7-5_646_898_552_587} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Write down the velocity of the insect when \(t = 0\).
  2. Show that the insect is instantaneously at rest when \(t = - 2\) and when \(t = 4\).
  3. Determine the velocity of the insect when its acceleration is zero. Write down the coordinates of the point A shown in Fig. 7.
  4. Calculate the distance travelled by the insect from \(t = 1\) to \(t = 4\).
  5. Write down the distance travelled by the insect in the time interval \(- 2 \leqslant t \leqslant 4\).
  6. How far does the insect walk in the time interval \(1 \leqslant t \leqslant 5\) ?
OCR MEI M1 2007 June Q8
8 A ball is kicked from ground level over horizontal ground. It leaves the ground at a speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) 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 2008 June Q1
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]{170edb27-324e-44df-8dc1-7d8fbad680fe-2_310_261_488_941} \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]{170edb27-324e-44df-8dc1-7d8fbad680fe-2_305_490_1354_829} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
    \end{figure}
  3. Calculate the new tension in the wire.
OCR MEI M1 2008 June Q2
2 A particle has a position vector \(\mathbf { r }\), where \(\mathbf { r } = 4 \mathbf { i } - 5 \mathbf { j }\) and \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors in the directions east and north respectively.
  1. Sketch \(\mathbf { r }\) on a diagram showing \(\mathbf { i }\) and \(\mathbf { j }\) and the origin O .
  2. Calculate the magnitude of \(\mathbf { r }\) and its direction as a bearing.
  3. Write down the vector that has the same direction as \(\mathbf { r }\) and three times its magnitude.
OCR MEI M1 2008 June Q3
3 An object of mass 5 kg has a constant acceleration of \(\binom { - 1 } { 2 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for \(0 \leqslant t \leqslant 4\), where \(t\) is the time in seconds.
  1. Calculate the force acting on the object. When \(t = 0\), the object has position vector \(\binom { - 2 } { 3 } \mathrm {~m}\) and velocity \(\binom { 4 } { 5 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the position vector of the object when \(t = 4\).
OCR MEI M1 2008 June Q4
4 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{170edb27-324e-44df-8dc1-7d8fbad680fe-3_346_981_781_584} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} Particles P and Q move in the same straight line. Particle P starts from rest and has a constant acceleration towards Q of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Particle Q starts 125 m from P at the same time and has a constant speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) away from P . The initial values are shown in Fig. 4.
  1. Write down expressions for the distances travelled by P and by Q at time \(t\) seconds after the start of the motion.
  2. How much time does it take for P to catch up with Q and how far does P travel in this time?
OCR MEI M1 2008 June Q5
5 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]{170edb27-324e-44df-8dc1-7d8fbad680fe-3_91_913_1959_616} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure} Calculate the tension in the wire joining the boxes.
OCR MEI M1 2008 June Q6
6 In this question take \(\boldsymbol { g } = \mathbf { 1 0 }\).
A golf ball is hit from ground level over horizontal ground. The initial velocity of the ball is \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.6\) and \(\cos \alpha = 0.8\). Air resistance may be neglected.
  1. Find an expression for the height of the ball above the ground \(t\) seconds after projection.
  2. Calculate the horizontal range of the ball. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{170edb27-324e-44df-8dc1-7d8fbad680fe-4_358_447_360_849} \captionsetup{labelformat=empty} \caption{Fig. 7.1}
    \end{figure} A box of mass 8 kg is supported by a continuous light string ACB that is fixed at A and at B and passes through a smooth ring on the box at C, as shown in Fig. 7.1. The box is in equilibrium and the tension in the string section AC is 60 N .
OCR MEI M1 2008 June Q8
8 The displacement, \(x \mathrm {~m}\), from the origin O of a particle on the \(x\)-axis is given by $$x = 10 + 36 t + 3 t ^ { 2 } - 2 t ^ { 3 }$$ where \(t\) is the time in seconds and \(- 4 \leqslant t \leqslant 6\).
  1. Write down the displacement of the particle when \(t = 0\).
  2. Find an expression in terms of \(t\) for the velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of the particle.
  3. Find an expression in terms of \(t\) for the acceleration of the particle.
  4. Find the maximum value of \(v\) in the interval \(- 4 \leqslant t \leqslant 6\).
  5. Show that \(v = 0\) only when \(t = - 2\) and when \(t = 3\). Find the values of \(x\) at these times.
  6. Calculate the distance travelled by the particle from \(t = 0\) to \(t = 4\).
  7. Determine how many times the particle passes through O in the interval \(- 4 \leqslant t \leqslant 6\).