Questions — OCR MEI (4301 questions)

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OCR MEI M1 Q6
19 marks Standard +0.3
6 A toy boat moves in a horizontal plane with position vector \(\mathbf { r } = x \mathbf { i } + y \mathbf { j }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are the standard unit vectors east and north respectively. The origin of the position vectors is at O . The displacements \(x\) and \(y\) are in metres. First consider only the motion of the boat parallel to the \(x\)-axis. For this motion $$x = 8 t - 2 t ^ { 2 }$$ The velocity of the boat in the \(x\)-direction is \(v _ { x } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find an expression in terms of \(t\) for \(v _ { x }\) and determine when the boat instantaneously has zero speed in the \(x\)-direction. Now consider only the motion of the boat parallel to the \(y\)-axis. For this motion $$v _ { y } = ( t - 2 ) ( 3 t - 2 )$$ where \(v _ { y } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of the boat in the \(y\)-direction at time \(t\) seconds.
  2. Given that \(y = 3\) when \(t = 1\), use integration to show that \(y = t ^ { 3 } - 4 t ^ { 2 } + 4 t + 2\). The position vector of the boat is given in terms of \(t\) by \(\mathbf { r } = \left( 8 t - 2 t ^ { 2 } \right) \mathbf { i } + \left( t ^ { 3 } - 4 t ^ { 2 } + 4 t + 2 \right) \mathbf { j }\).
  3. Find the time(s) when the boat is due north of O and also the distance of the boat from O at any such times.
  4. Find the time(s) when the boat is instantaneously at rest. Find the distance of the boat from O at any such times.
  5. Plot a graph of the path of the boat for \(0 \leqslant t \leqslant 2\).
OCR MEI M1 Q1
7 marks Moderate -0.5
1 The velocity of a model boat, \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { 1 }\), is given by $$\mathbf { v } = \binom { 5 } { 10 } + t \binom { 6 } { 8 }$$ where \(t\) is the time in seconds and the vectors \(\binom { 1 } { 0 }\) and \(\binom { 0 } { 1 }\) are east and north respectively.
  1. Show that when \(t = 2.5\) the boat is travelling south-east (i.e. on a bearing of \(135 ^ { \circ }\) ). Calculate its speed at this time. The boat is at a point O when \(t = 0\).
  2. Calculate the bearing of the boat from O when \(t = 2.5\).
OCR MEI M1 Q2
6 marks Moderate -0.5
2 The acceleration of a particle of mass 4 kg is given by \(\mathbf { a } = ( 9 \mathbf { i } - 4 t \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { 2 }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors and \(t\) is the time in seconds.
  1. Find the acceleration of the particle when \(t = 0\) and also when \(t = 3\).
  2. Calculate the force acting on the particle when \(t = 3\). The particle has velocity \(( 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } { } ^ { 1 }\) when \(t = 1\).
  3. Find an expression for the velocity of the particle at time \(t\).
OCR MEI M1 Q3
7 marks Moderate -0.3
3 The position vector, \(r\), of a particle of mass 4 kg at time \(t\) is given by $$\mathbf { r } = t ^ { 2 } \mathbf { i } + \left( 5 t - 2 t ^ { 2 } \right) \mathbf { j }$$ where \(\mathbf { i }\) and \(\mathbf { j }\) are the standard unit vectors, lengths are in metres and time is in seconds.
  1. Find an expression for the acceleration of the particle. The particle is subject to a force \(\mathbf { F }\) and a force \(12 \mathbf { j } \mathbf { N }\).
  2. Find \(\mathbf { F }\).
OCR MEI M1 Q4
18 marks Moderate -0.3
4 A ring is moving on a straight wire. Its velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds after passing a point Q .
Model A for the motion of the ring gives the velocity-time graph for \(0 \leqslant t \leqslant 6\) shown in Fig. 7 . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{196bd74f-c2b2-4cb3-b03c-8ecd9fce9c11-2_937_1414_325_404} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure} Use model A to calculate the following.
  1. The acceleration of the ring when \(t = 0.5\).
  2. The displacement of the ring from Q when
    (A) \(t = 2\),
    (B) \(t = 6\). In an alternative model B , the velocity of the ring is given by \(v = 2 t ^ { 2 } - 14 t + 20\) for \(0 \leqslant t \leqslant 6\).
  3. Calculate the acceleration of the ring at \(t = 0.5\) as given by model B.
  4. Calculate by how much the models differ in their values for the least \(v\) in the time interval \(0 \leqslant t \leqslant 6\).
  5. Calculate the displacement of the ring from Q when \(t = 6\) as given by model B .
OCR MEI M1 Q1
8 marks Moderate -0.8
1 Force \(\mathbf { F } _ { 1 }\) is \(\binom { 6 } { 13 } \mathrm {~N}\) and force \(\mathbf { F } _ { 2 }\) is \(\binom { 3 } { 5 }\), where \({ } _ { 0 }\) 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 Q2
18 marks Moderate -0.3
2 The speed of a 100 metre runner in \(\mathrm { m } \mathrm { s } ^ { - 1 }\) 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]{4f80ea36-001f-4a00-849f-542f5072516b-2_1022_1503_524_290} \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, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) 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? In this question take \(g\) as \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
    A small ball is released from rest. It falls for 2 seconds and is then brought to rest over the next 5 seconds. This motion is modelled in the speed-time graph Fig. 6. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{4f80ea36-001f-4a00-849f-542f5072516b-3_658_1101_281_503} \captionsetup{labelformat=empty} \caption{Fig. 6}
    \end{figure} For this model,
  5. calculate the distance fallen from \(t = 0\) to \(t = 7\),
  6. find the acceleration of the ball from \(t = 2\) to \(t = 6\), specifying the direction,
  7. obtain an expression in terms of \(t\) for the downward speed of the ball from \(t = 2\) to \(t = 6\),
  8. state the assumption that has been made about the resistance to motion from \(t = 0\) to \(t = 2\). The part of the motion from \(t = 2\) to \(t = 7\) is now modelled by \(v = - \frac { 3 } { 2 } t ^ { 2 } + \frac { 19 } { 2 } t + 7\).
  9. Verify that \(v\) agrees with the values given in Fig, 6 at \(t = 2 , t = 6\) and \(t = 7\).
  10. Calculate the distance fallen from \(t = 2\) to \(t = 7\) according to this model.
OCR MEI M1 Q2
21 marks Standard +0.3
2 A box of mass 8 kg slides on a horizontal table against a constant resistance of 11.2 N .
  1. What horizontal force is applied to the box if it is sliding with acceleration of magnitude \(2 \mathrm {~ms} ^ { - 2 }\) ? Fig. 7 shows the box of mass 8 kg on a long, rough, horizontal table. A sphere of mass 6 kg is attached to the box by means of a light inextensible string that passes over a smooth pulley. The section of the string between the pulley and the box is parallel to the table. The constant frictional force of 11.2 N opposes the motion of the box. A force of 105 N parallel to the table acts on the box in the direction shown, and the acceleration of the system is in that direction. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0fbef619-ad15-4e46-be35-e17fed9952c0-2_372_878_870_683} \captionsetup{labelformat=empty} \caption{Fig. 7}
    \end{figure}
  2. What information in the question indicates that while the string is taut the box and sphere have the same acceleration?
  3. Draw two separate diagrams, one showing all the horizontal forces acting on the box and the other showing all the forces acting on the sphere.
  4. Show that the magnitude of the acceleration of the system is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and find the tension in the string. The system is stationary when the sphere is at point P . When the sphere is 1.8 m above P the string breaks, leaving the sphere moving upwards at a speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  5. (A) Write down the value of the acceleration of the sphere after the string breaks.
    (B) The sphere passes through P again at time \(T\) seconds after the string breaks. Show that \(T\) is the positive root of the equation \(4.9 T ^ { 2 } - 3 T - 1.8 = 0\).
    ( \(C\) ) Using part ( \(B\) ), or otherwise, calculate the total time that elapses after the sphere moves from P before the sphere again passes through P .
OCR MEI M1 Q2
18 marks Standard +0.3
2 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 Q3
16 marks Moderate -0.3
3 A trolley C of mass 8 kg with rusty axle bearings is initially at rest on a horizontal floor.
The trolley stays at rest when it is pulled by a horizontal string with tension 25 N , as shown in Fig. 8.1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f5f9b9b7-6766-4f8e-b011-506051104123-3_249_1096_314_558} \captionsetup{labelformat=empty} \caption{Fig. 8.1}
\end{figure}
  1. State the magnitude of the horizontal resistance opposing the pull. A second trolley D of mass 10 kg is connected to trolley C by means of a light, horizontal rod.
    The string now has tension 50 N , and is at an angle of \(25 ^ { \circ }\) to the horizontal, as shown in Fig. 8.2. The two trolleys stay at rest. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f5f9b9b7-6766-4f8e-b011-506051104123-3_297_1180_971_741} \captionsetup{labelformat=empty} \caption{Fig. 8.2}
    \end{figure}
  2. Calculate the magnitude of the total horizontal resistance acting on the two trolleys opposing the pull.
  3. Calculate the normal reaction of the floor on trolley C . The axle bearings of the trolleys are oiled and the total horizontal resistance to the motion of the two trolleys is now 20 N . The two trolleys are still pulled by the string with tension 50 N , as shown in Fig. 8.2.
  4. Calculate the acceleration of the trolleys. In a new situation, the trolleys are on a slope at \(5 ^ { \circ }\) to the horizontal and are initially travelling down the slope at \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistances are 15 N to the motion of D and 5 N to the motion of C . There is no string attached. The rod connecting the trolleys is parallel to the slope. This situation is shown in Fig. 8.3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f5f9b9b7-6766-4f8e-b011-506051104123-3_351_1285_2038_466} \captionsetup{labelformat=empty} \caption{Fig. 8.3}
    \end{figure}
  5. Calculate the speed of the trolleys after 2 seconds and also the force in the rod connecting the PhysicsAptMaths, statter \&REther this rod is in tension or thrust (compression).
OCR MEI M1 Q1
6 marks Moderate -0.8
1 Fig. 1 shows the speed-time graph of a runner during part of his training. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{569e7c0e-7c33-47c9-b986-8587ea239f0a-1_1068_1586_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 Q2
18 marks Standard +0.3
2 A train consists of a locomotive pulling 17 identical trucks.
The mass of the locomotive is 120 tonnes and the mass of each truck is 40 tonnes. The locomotive gives a driving force of 121000 N . The resistance to motion on each truck is \(R \mathrm {~N}\) and the resistance on the locomotive is \(5 R \mathrm {~N}\).
Initially the train is travelling on a straight horizontal track and its acceleration is \(0.11 \mathrm {~ms} ^ { - 2 }\).
  1. Show that \(R = 1500\).
  2. Find the tensions in the couplings between
    (A) the last two trucks,
    (B) the locomotive and the first truck. The train now comes to a place where the track goes up a straight, uniform slope at an angle \(\alpha\) with the horizontal, where \(\sin \alpha = \frac { 1 } { 80 }\). The driving force and the resistance forces remain the same as before.
  3. Find the magnitude and direction of the acceleration of the train. The train then comes to a straight uniform downward slope at an angle \(\beta\) to the horizontal.
    The driver of the train reduces the driving force to zero and the resistance forces remain the same as before. The train then travels at a constant speed down the slope.
  4. Find the value of \(\beta\).
OCR MEI M1 Q3
16 marks Moderate -0.3
3 A point P on a piece of machinery is moving in a vertical straight line. The displacement of P above ground level at time \(t\) seconds is \(y\) metres. The displacement-time graph for the motion during the time interval \(0 \leqslant t \leqslant 4\) is shown in Fig. 7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{569e7c0e-7c33-47c9-b986-8587ea239f0a-3_1020_1333_352_439} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Using the graph, determine for the time interval \(0 \leqslant t \leqslant 4\)
    (A) the greatest displacement of P above its position when \(t = 0\),
    (B) the greatest distance of P from its position when \(t = 0\),
    (C) the time interval in which P is moving downwards,
    (D) the times when P is instantaneously at rest. The displacement of P in the time interval \(0 \leqslant t \leqslant 3\) is given by \(y = - 4 t ^ { 2 } + 8 t + 12\).
  2. Use calculus to find expressions in terms of \(t\) for the velocity and for the acceleration of P in the interval \(0 \leqslant t \leqslant 3\).
  3. At what times does P have a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the interval \(0 \leqslant t \leqslant 3\) ? In the time interval \(3 \leqslant t \leqslant 4 , \mathrm { P }\) has a constant acceleration of \(32 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). There is no sudden change in velocity when \(t = 3\).
  4. Find an expression in terms of \(t\) for the displacement of P in the interval \(3 \leqslant t \leqslant 4\).
OCR MEI M1 Q4
6 marks Moderate -0.8
4 The velocity-time graph shown in Fig. 1 represents the straight line motion of a toy car. All the lines on the graph are straight. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{569e7c0e-7c33-47c9-b986-8587ea239f0a-4_579_1319_381_449} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} The car starts at the point A at \(t = 0\) and in the next 8 seconds moves to a point B .
  1. Find the distance from A to B .
    \(T\) seconds after leaving A , the car is at a point C which is a distance of 10 m from B .
  2. Find the value of \(T\).
  3. Find the displacement from A to C .
OCR MEI M1 Q5
8 marks Moderate -0.8
5 Fig. 3 is a sketch of the velocity-time graph modelling the velocity of a sprinter at the start of a race. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{569e7c0e-7c33-47c9-b986-8587ea239f0a-5_575_1086_482_551} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
  1. How can you tell from the sketch that the acceleration is not modelled as being constant for \(0 \leqslant t \leqslant 4\) ? The velocity of the sprinter, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), for the time interval \(0 \leqslant t \leqslant 4\) is modelled by the expression $$v = 3 t - \frac { 3 } { 8 } t ^ { 2 } .$$
  2. Find the acceleration that the model predicts for \(t = 4\) and comment on what this suggests about the running of the sprinter.
  3. Calculate the distance run by the sprinter from \(t = 1\) to \(t = 4\).
OCR MEI M1 Q6
8 marks Moderate -0.3
6 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 \(10 \mathrm {~ms} { } ^ { 1 }\) to \(30 \mathrm {~ms} { } ^ { 1 }\) 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 Q1
5 marks Moderate -0.8
1 A ring is moving up and down a vertical pole. The displacement, \(s \mathrm {~m}\), of the ring above a mark on the pole is modelled by the displacement-time graph shown in Fig. 1. The three sections of the graph are straight lines. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bdbebc7f-0cb1-4203-8058-7614ba291508-1_763_1057_439_580} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure}
  1. Calculate the velocity of the ring in the interval \(0 < t < 2\) and in the interval \(2 < t < 3.5\).
  2. Sketch a velocity-time graph for the motion of the ring during the 4 seconds.
  3. State the direction of motion of the ring when
    (A) \(t = 1\),
    (B) \(t = 2.75\),
    (C) \(t = 3.25\).
OCR MEI M1 Q2
4 marks Moderate -0.3
2 Fig. 2 shows an acceleration-time graph modelling the motion of a particle. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bdbebc7f-0cb1-4203-8058-7614ba291508-2_684_1068_408_586} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} At \(t = 0\) the particle has a velocity of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive direction.
  1. Find the velocity of the particle when \(t = 2\).
  2. At what time is the particle travelling in the negative direction with a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) ?
OCR MEI M1 Q3
6 marks Easy -1.2
3 A cyclist starts from rest and takes 10 seconds to accelerate at a constant rate up to a speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After travelling at this speed for 20 seconds, the cyclist then decelerates to rest at a constant rate over the next 5 seconds.
  1. Sketch a velocity-time graph for the motion.
  2. Calculate the distance travelled by the cyclist.
OCR MEI M1 Q4
4 marks Moderate -0.5
4 Fig. 1 is the velocity-time graph for the motion of a body. The velocity of the body is \(v \mathrm {~m} \mathrm {~s} { } ^ { 1 }\) at time \(t\) seconds. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bdbebc7f-0cb1-4203-8058-7614ba291508-3_656_1344_401_399} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} The displacement of the body from \(t = 0\) to \(t = 100\) is 1400 m . Find the value of \(V\).
OCR MEI M1 Q2
4 marks Moderate -0.5
2 Fig. 1 is the velocity-time graph for the motion of a body. The velocity of the body is \(v \mathrm {~m} \mathrm {~s} { } ^ { 1 }\) at time \(t\) seconds. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{578555c7-e316-47d3-876a-0b6accce8946-1_662_1354_915_441} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} The displacement of the body from \(t = 0\) to \(t = 100\) is 1400 m . Find the value of \(V\).
OCR MEI M1 Q3
6 marks Moderate -0.3
3 A particle travels in a straight line during the time interval \(0 \leqslant t \leqslant 12\), where \(t\) is the time in seconds. Fig. 1 is the velocity-time graph for the motion. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{578555c7-e316-47d3-876a-0b6accce8946-2_445_854_426_667} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure}
  1. Calculate the acceleration of the particle in the interval \(0 < t < 6\).
  2. Calculate the distance travelled by the particle from \(t = 0\) to \(t = 4\).
  3. When \(t = 0\) the particle is at A . Calculate how close the particle gets to A during the interval \(4 \leqslant t \leqslant 12\). In this question take \(\boldsymbol { g }\) as \(\mathbf { 1 0 } \mathrm { m } \mathrm { s } ^ { \mathbf { 2 } }\).
    A small ball is released from rest. It falls for 2 seconds and is then brought to rest over the next 5 seconds. This motion is modelled in the speed-time graph Fig. 6. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{578555c7-e316-47d3-876a-0b6accce8946-3_596_1004_517_499} \captionsetup{labelformat=empty} \caption{Fig. 6}
    \end{figure} For this model,
  4. calculate the distance fallen from \(t = 0\) to \(t = 7\),
  5. find the acceleration of the ball from \(t = 2\) to \(t = 6\), specifying the direction,
  6. obtain an expression in terms of \(t\) for the downward speed of the ball from \(t = 2\) to \(t = 6\),
  7. state the assumption that has been made about the resistance to motion from \(t = 0\) to \(t = 2\). The part of the motion from \(t = 2\) to \(t = 7\) is now modelled by \(v = - \frac { 3 } { 2 } t ^ { 2 } + \frac { 19 } { 2 } t + 7\).
  8. Verify that \(v\) agrees with the values given in Fig. 6 at \(t = 2 , t = 6\) and \(t = 7\).
  9. Calculate the distance fallen from \(t = 2\) to \(t = 7\) according to this model.
OCR MEI M1 Q5
18 marks Standard +0.3
5 A box of emergency supplies is dropped to victims of a natural disaster from a stationary helicopter at a height of 1000 metres. The initial velocity of the box is zero. At time \(t \mathrm {~s}\) after being dropped, the acceleration, \(a \mathrm {~ms} ^ { - 2 }\), of the box in the vertically downwards direction is modelled by $$\begin{aligned} & a = 10 - t \text { for } 0 \leqslant t \leqslant 10 \\ & a = 0 \quad \text { for } \quad t > 10 \end{aligned}$$
  1. Find an expression for the velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of the box in the vertically downwards direction in terms of \(t\) for \(0 \leqslant t \leqslant 10\). Show that for \(t > 10 , v = 50\).
  2. Draw a sketch graph of \(v\) against \(t\) for \(0 \leqslant t \leqslant 20\).
  3. Show that the height, \(h \mathrm {~m}\), of the box above the ground at time \(t \mathrm {~s}\) is given, for \(0 \leqslant t \leqslant 10\), by $$h = 1000 - 5 t ^ { 2 } + \frac { 1 } { 6 } t ^ { 3 }$$ Find the height of the box when \(t = 10\).
  4. Find the value of \(t\) when the box hits the ground.
  5. Some of the supplies in the box are damaged when the box hits the ground. So measures are considered to reduce the speed with which the box hits the ground the next time one is dropped. Two different proposals are made. Carry out suitable calculations and then comment on each of them.
    (A) The box should be dropped from a height of 500 m instead of 1000 m .
    (B) The box should be fitted with a parachute so that its acceleration is given by $$\begin{gathered} \quad a = 10 - 2 t \text { for } 0 \leqslant t \leqslant 5 , \\ a = 0 \quad \text { for } \quad t > 5 . \end{gathered}$$ \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{578555c7-e316-47d3-876a-0b6accce8946-5_342_979_319_633} \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.
  6. Write down expressions for the distances travelled by P and by Q at time \(t\) seconds after the start of the motion.
  7. 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 Q1
8 marks Standard +0.3
1 Fig. 2 shows a 6 kg block on a smooth horizontal table. It is connected to blocks of mass 2 kg and 9 kg by two light strings which pass over smooth pulleys at the edges of the table. The parts of the strings attached to the 6 kg block are horizontal. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9bff41e0-7be0-4932-ae50-a612abb3fe19-1_345_1141_364_480} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure}
  1. Draw three separate diagrams showing all the forces acting on each of the blocks.
  2. Calculate the acceleration of the system and the tension in each string.
OCR MEI M1 Q2
18 marks Standard +0.3
2 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]{9bff41e0-7be0-4932-ae50-a612abb3fe19-2_239_1137_636_484} \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]{9bff41e0-7be0-4932-ae50-a612abb3fe19-2_216_425_1964_870} \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 \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) they can get the engine to start.