Questions M1 (2067 questions)

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CAIE M1 2019 June Q4
10 marks Standard +0.8
  1. Show that, before the string breaks, the magnitude of the acceleration of each particle is \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and find the tension in the string.
  2. Find the difference in the times that it takes the particles to hit the ground.
CAIE M1 2011 November Q5
8 marks Standard +0.8
  1. Show that \(\mu \geqslant \frac { 6 } { 17 }\). When the applied force acts upwards as in Fig. 2 the block slides along the floor.
  2. Find another inequality for \(\mu\).
CAIE M1 2012 November Q5
8 marks Standard +0.3
  1. Find the value of \(\theta\). At time 4.8 s after leaving \(A\), the particle comes to rest at \(C\).
  2. Find the coefficient of friction between \(P\) and the rough part of the plane.
CAIE M1 2014 November Q6
9 marks Standard +0.3
  1. the work done against the frictional force acting on \(B\),
  2. the loss of potential energy of the system,
  3. the gain in kinetic energy of the system. At the instant when \(B\) has moved 0.9 m the string breaks. \(A\) is at a height of 0.54 m above a horizontal floor at this instant.
    (ii) Find the speed with which \(A\) reaches the floor. \(6 \quad A B C\) is a line of greatest slope of a plane inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.28\) and \(\cos \alpha = 0.96\). The point \(A\) is at the top of the plane, the point \(C\) is at the bottom of the plane and the length of \(A C\) is 5 m . The part of the plane above the level of \(B\) is smooth and the part below the level of \(B\) is rough. A particle \(P\) is released from rest at \(A\) and reaches \(C\) with a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of friction between \(P\) and the part of the plane below \(B\) is 0.5 . Find
    1. the acceleration of \(P\) while moving
      1. from \(A\) to \(B\),
      2. from \(B\) to \(C\),
      3. the distance \(A B\),
      4. the time taken for \(P\) to move from \(A\) to \(C\).
OCR M1 2010 January Q5
11 marks Moderate -0.3
  1. Find the value of \(t\) when \(A\) and \(B\) have the same speed.
  2. Calculate the value of \(t\) when \(B\) overtakes \(A\).
  3. On a single diagram, sketch the \(( t , x )\) graphs for the two cyclists for the time from \(t = 0\) until after \(B\) has overtaken \(A\).
OCR M1 2015 June Q3
8 marks Standard +0.3
  1. Calculate the distance \(A\) cycles, and hence find the period of time for which \(B\) walks before finding the bicycle.
  2. Find \(T\).
  3. Calculate the distance \(A\) and \(B\) each travel.
OCR MEI M1 2008 June Q7
17 marks Moderate -0.3
  1. What information in the question indicates that the tension in the string section CB is also 60 N ?
  2. Show that the string sections AC and CB are equally inclined to the horizontal (so that \(\alpha = \beta\) in Fig. 7.1).
  3. Calculate the angle of the string sections AC and CB to the horizontal. In a different situation the same box is supported by two separate light strings, PC and QC, that are tied to the box at C . There is also a horizontal force of 10 N acting at C . This force and the angles between these strings and the horizontal are shown in Fig. 7.2. The box is in equilibrium. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{170edb27-324e-44df-8dc1-7d8fbad680fe-4_323_503_1649_822} \captionsetup{labelformat=empty} \caption{Fig. 7.2}
    \end{figure}
  4. Calculate the tensions in the two strings.
OCR MEI M1 2013 June Q7
18 marks Moderate -0.3
  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]{83e69140-4abf-4713-85da-922ce7530e47-5_383_360_534_854} \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 2005 January Q6
19 marks Moderate -0.8
6 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]{c84a748a-a6f4-48c5-b864-fe543569bdf5-5_659_1105_578_493} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure} For this model,
  1. calculate the distance fallen from \(t = 0\) to \(t = 7\),
  2. find the acceleration of the ball from \(t = 2\) to \(t = 6\), specifying the direction,
  3. obtain an expression in terms of \(t\) for the downward speed of the ball from \(t = 2\) to \(t = 6\),
  4. 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 \(v = - \frac { 3 } { 2 } t ^ { 2 } + \frac { 19 } { 2 } t + 7\).
  5. Verify that \(v\) agrees with the values given in Fig. 6 at \(t = 2 , t = 6\) and \(t = 7\).
  6. Calculate the distance fallen from \(t = 2\) to \(t = 7\) according to this model.
OCR MEI M1 2007 January Q8
18 marks Standard +0.3
8 In this question the value of \(\boldsymbol { g \) should be taken as \(\mathbf { 1 0 } \mathbf { m ~ s } ^ { \mathbf { - 2 } }\).} As shown in Fig. 8, particles A and B are projected towards one another. Each particle has an initial speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically and \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) horizontally. Initially A and B are 70 m apart horizontally and B is 15 m higher than A . Both particles are projected over horizontal ground. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{52d6c914-b204-4587-a82e-fbab6693fcf8-6_476_1111_518_475} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Show that, \(t\) seconds after projection, the height in metres of each particle above its point of projection is \(10 t - 5 t ^ { 2 }\).
  2. Calculate the horizontal range of A . Deduce that A hits the horizontal ground between the initial positions of A and B .
  3. Calculate the horizontal distance travelled by B before reaching the ground.
  4. Show that the paths of the particles cross but that the particles do not collide if they are projected at the same time. In fact, particle A is projected 2 seconds after particle B .
  5. Verify that the particles collide 0.75 seconds after A is projected.
OCR MEI M1 2016 June Q6
18 marks Moderate -0.3
6 In this question you should take \(\boldsymbol { g \) to be \(\mathbf { 1 0 } \mathrm { ms } ^ { \boldsymbol { - } \mathbf { 2 } }\).} Piran finds a disused mineshaft on his land and wants to know its depth, \(d\) metres.
Local records state that the mineshaft is between 150 and 200 metres deep.
He drops a small stone down the mineshaft and records the time, \(T\) seconds, until he hears it hit the bottom. It takes 8.0 seconds. Piran tries three models, \(\mathrm { A } , \mathrm { B }\) and C .
In model A, Piran uses the formula \(d = 5 T ^ { 2 }\) to estimate the depth.
  1. Find the depth that model A gives and comment on whether it is consistent with the local records. Explain how the formula in model A is obtained. In model B, Piran uses the speed-time graph in Fig. 6. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{4c8c96cf-5184-46e4-9c45-a8a80d0a6ff8-5_762_1176_1087_424} \captionsetup{labelformat=empty} \caption{Fig. 6}
    \end{figure}
  2. Calculate the depth of the mineshaft according to model B. Comment on whether this depth is consistent with the local records.
  3. Describe briefly one respect in which model B is the same as model A and one respect in which it is different. Piran then tries model C in which the speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by $$\begin{aligned} & v = 10 t - t ^ { 2 } \text { for } 0 \leqslant t \leqslant 5 \\ & v = 25 \text { for } 5 < t \leqslant 8 \end{aligned}$$
  4. Calculate the depth of the mineshaft according to model C. Comment on whether this depth is consistent with the local records.
  5. Describe briefly one respect in which model C is similar to model B and one respect in which it is different.
OCR MEI M1 Q1
18 marks Standard +0.3
1 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 Q2
18 marks Standard +0.3
2 In this question the value of \(g\) should be taken as \(10 \mathrm {~m \mathrm {~s} ^ { 2 }\).} As shown in Fig. 8, particles A and B are projected towards one another. Each particle has an initial speed of \(10 \mathrm {~m} \mathrm {~s} ^ { 1 }\) vertically and \(20 \mathrm {~m} \mathrm {~s} { } ^ { 1 }\) horizontally. Initially A and B are 70 m apart horizontally and B is 15 m higher than A . Both particles are projected over horizontal ground. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{362d5995-bd39-4b07-b6a4-63eb1dd3e69d-2_461_1114_464_505} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Show that, \(t\) seconds after projection, the height in metres of each particle above its point of projection is \(10 t - 5 t ^ { 2 }\).
  2. Calculate the horizontal range of A . Deduce that A hits the horizontal ground between the initial positions of A and B .
  3. Calculate the horizontal distance travelled by B before reaching the ground.
  4. Show that the paths of the particles cross but that the particles do not collide if they are projected at the same time. In fact, particle A is projected 2 seconds after particle B .
  5. Verify that the particles collide 0.75 seconds after A is projected.
Edexcel M1 Q3
Moderate -0.8
3. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-005_851_1073_312_456}
\end{figure} A sprinter runs a race of 200 m . Her total time for running the race is 25 s . Figure 2 is a sketch of the speed-time graph for the motion of the sprinter. She starts from rest and accelerates uniformly to a speed of \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 4 s . The speed of \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is maintained for 16 s and she then decelerates uniformly to a speed of \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the end of the race. Calculate
  1. the distance covered by the sprinter in the first 20 s of the race,
  2. the value of \(u\),
  3. the deceleration of the sprinter in the last 5 s of the race.
Edexcel M1 Q4
Moderate -0.8
4. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-007_330_675_287_644}
\end{figure} A particle \(P\) of mass 2.5 kg rests in equilibrium on a rough plane under the action of a force of magnitude \(X\) newtons acting up a line of greatest slope of the plane, as shown in Figure 3. The plane is inclined at \(20 ^ { \circ }\) to the horizontal. The coefficient of friction between \(P\) and the plane is 0.4 . The particle is in limiting equilibrium and is on the point of moving up the plane. Calculate
  1. the normal reaction of the plane on \(P\),
  2. the value of \(X\). The force of magnitude \(X\) newtons is now removed.
  3. Show that \(P\) remains in equilibrium on the plane.
Edexcel M1 Q5
Standard +0.3
5. Figure 4 \includegraphics[max width=\textwidth, alt={}, center]{94d9432d-1723-4549-ad5e-d4be0f5fd083-009_609_1026_301_516} A block of wood \(A\) of mass 0.5 kg rests on a rough horizontal table and is attached to one end of a light inextensible string. The string passes over a small smooth pulley \(P\) fixed at the edge of the table. The other end of the string is attached to a ball \(B\) of mass 0.8 kg which hangs freely below the pulley, as shown in Figure 4. The coefficient of friction between \(A\) and the table is \(\mu\). The system is released from rest with the string taut. After release, \(B\) descends a distance of 0.4 m in 0.5 s . Modelling \(A\) and \(B\) as particles, calculate
  1. the acceleration of \(B\),
  2. the tension in the string,
  3. the value of \(\mu\).
  4. State how in your calculations you have used the information that the string is inextensible.
Edexcel M1 Q7
Moderate -0.3
7. Two ships \(P\) and \(Q\) are travelling at night with constant velocities. At midnight, \(P\) is at the point with position vector \(( 20 \mathbf { i } + 10 \mathbf { j } ) \mathrm { km }\) relative to a fixed origin \(O\). At the same time, \(Q\) is at the point with position vector \(( 14 \mathbf { i } - 6 \mathbf { j } ) \mathrm { km }\). Three hours later, \(P\) is at the point with position vector \(( 29 \mathbf { i } + 34 \mathbf { j } ) \mathrm { km }\). The ship \(Q\) travels with velocity \(12 \mathbf { j } \mathrm {~km} \mathrm {~h} ^ { - 1 }\). At time \(t\) hours after midnight, the position vectors of \(P\) and \(Q\) are \(\mathbf { p } \mathrm { km }\) and \(\mathbf { q } \mathrm { km }\) respectively. Find
  1. the velocity of \(P\), in terms of \(\mathbf { i }\) and \(\mathbf { j }\),
  2. expressions for \(\mathbf { p }\) and \(\mathbf { q }\), in terms of \(t\), i and \(\mathbf { j }\). At time \(t\) hours after midnight, the distance between \(P\) and \(Q\) is \(d \mathrm {~km}\).
  3. By finding an expression for \(\overrightarrow { P Q }\), show that $$d ^ { 2 } = 25 t ^ { 2 } - 92 t + 292$$ Weather conditions are such that an observer on \(P\) can only see the lights on \(Q\) when the distance between \(P\) and \(Q\) is 15 km or less. Given that when \(t = 1\), the lights on \(Q\) move into sight of the observer,
  4. find the time, to the nearest minute, at which the lights on \(Q\) move out of sight of the observer.
    1. In taking off, an aircraft moves on a straight runway \(A B\) of length 1.2 km . The aircraft moves from \(A\) with initial speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It moves with constant acceleration and 20 s later it leaves the runway at \(C\) with speed \(74 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
      1. the acceleration of the aircraft,
      2. the distance \(B C\).
      3. Two small steel balls \(A\) and \(B\) have mass 0.6 kg and 0.2 kg respectively. They are moving towards each other in opposite directions on a smooth horizontal table when they collide directly. Immediately before the collision, the speed of \(A\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after the collision, the direction of motion of \(A\) is unchanged and the speed of \(B\) is twice the speed of \(A\). Find
    2. the speed of \(A\) immediately after the collision,
    3. the magnitude of the impulse exerted on \(B\) in the collision.
    \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-018_282_707_278_699}
    \end{figure}
Edexcel M1 Q8
Moderate -0.3
  1. \hspace{0pt} [In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal vectors due east and north respectively.]
AQA M1 Q1
Moderate -0.8
1 A particle \(A\) moves across a smooth horizontal surface in a straight line. The particle \(A\) has mass 2 kg and speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A particle \(B\), which has mass 3 kg , is at rest on the surface. The particle \(A\) collides with the particle \(B\). \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-003_147_506_644_733}
  1. If, after the collision, \(A\) is at rest and \(B\) moves away from \(A\), find the speed of \(B\).
  2. If, after the collision, \(A\) and \(B\) move away from each other with speeds \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(4 v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively, as shown in the diagram below, find the value of \(v\). \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-003_138_506_1144_730}
AQA M1 Q4
Standard +0.3
4 Water flows in a constant direction at a constant speed of \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A boat travels in the water at a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to the water.
  1. The direction in which the boat travels relative to the water is perpendicular to the direction of motion of the water. The resultant velocity of the boat is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(74 ^ { \circ }\) to the direction of motion of the water, as shown in the diagram. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-004_120_164_662_488} \captionsetup{labelformat=empty} \caption{Velocity of the water}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-004_126_186_667_890} \captionsetup{labelformat=empty} \caption{Velocity of the boat relative to the water}
    \end{figure}
    1. Find \(V\).
    2. Show that \(u = 3.44\), correct to three significant figures.
  2. The boat changes course so that it travels relative to the water at an angle of \(45 ^ { \circ }\) to the direction of motion of the water. The resultant velocity of the boat is now of magnitude \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The velocity of the water is unchanged, as shown in the diagram below. $$\xrightarrow { 3.44 \mathrm {~m} \mathrm {~s} ^ { - 1 } }$$
    \includegraphics[max width=\textwidth, alt={}]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-004_132_273_1493_895}
    Velocity of the boat relative to the water \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-004_232_355_1498_1384} Find the value of \(v\).
    (4 marks)
AQA M1 Q5
Moderate -0.8
5 A golf ball is projected from a point \(O\) with initial velocity \(V\) at an angle \(\alpha\) to the horizontal. The ball first hits the ground at a point \(A\) which is at the same horizontal level as \(O\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-005_227_602_484_735} It is given that \(V \cos \alpha = 6 u\) and \(V \sin \alpha = 2.5 u\).
  1. Show that the time taken for the ball to travel from \(O\) to \(A\) is \(\frac { 5 u } { g }\).
  2. Find, in terms of \(g\) and \(u\), the distance \(O A\).
  3. Find \(V\), in terms of \(u\).
  4. State, in terms of \(u\), the least speed of the ball during its flight from \(O\) to \(A\).
AQA M1 Q6
Moderate -0.8
6 A van moves from rest on a straight horizontal road.
  1. In a simple model, the first 30 seconds of the motion are represented by three separate stages, each lasting 10 seconds and each with a constant acceleration. During the first stage, the van accelerates from rest to a velocity of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    During the second stage, the van accelerates from \(4 \mathrm {~ms} ^ { - 1 }\) to \(12 \mathrm {~ms} ^ { - 1 }\).
    During the third stage, the van accelerates from \(12 \mathrm {~ms} ^ { - 1 }\) to \(16 \mathrm {~ms} ^ { - 1 }\).
    1. Sketch a velocity-time graph to represent the motion of the van during the first 30 seconds of its motion.
    2. Find the total distance that the van travels during the 30 seconds.
    3. Find the average speed of the van during the 30 seconds.
    4. Find the greatest acceleration of the van during the 30 seconds.
  2. In another model of the 30 seconds of the motion, the acceleration of the van is assumed to vary during the first and third stages of the motion, but to be constant during the second stage, as shown in the velocity-time graph below. \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-006_554_1138_1432_539} The velocity of the van takes the same values at the beginning and the end of each stage of the motion as in part (a).
    1. State, with a reason, whether the distance travelled by the van during the first 10 seconds of the motion in this model is greater or less than the distance travelled during the same time interval in the model in part (a).
    2. Give one reason why this model represents the motion of the van more realistically than the model in part (a).
AQA M1 Q7
Moderate -0.8
7 A builder ties two identical buckets, \(P\) and \(Q\), to the ends of a light inextensible rope. He hangs the rope over a smooth beam so that the buckets hang in equilibrium, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-007_360_296_502_904} The buckets are each of mass 0.6 kg .
    1. State the magnitude of the tension in the rope.
    2. State the magnitude and direction of the force exerted on the beam by the rope.
  2. The bucket \(Q\) is held at rest while a stone, of mass 0.2 kg , is placed inside it. The system is then released from rest and, in the subsequent motion, bucket \(Q\) moves vertically downwards with the stone inside.
    1. By forming an equation of motion for each bucket, show that the magnitude of the tension in the rope during the motion is 6.72 newtons, correct to three significant figures.
    2. State the magnitude of the force exerted on the beam by the rope while the motion takes place.
AQA M1 2006 January Q1
6 marks Moderate -0.8
1 A particle \(A\) moves across a smooth horizontal surface in a straight line. The particle \(A\) has mass 2 kg and speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A particle \(B\), which has mass 3 kg , is at rest on the surface. The particle \(A\) collides with the particle \(B\). \includegraphics[max width=\textwidth, alt={}, center]{c220e6c4-2676-4022-8301-7d720dc082b2-2_147_506_644_733}
  1. If, after the collision, \(A\) is at rest and \(B\) moves away from \(A\), find the speed of \(B\).
  2. If, after the collision, \(A\) and \(B\) move away from each other with speeds \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(4 v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively, as shown in the diagram below, find the value of \(v\). \includegraphics[max width=\textwidth, alt={}, center]{c220e6c4-2676-4022-8301-7d720dc082b2-2_138_506_1144_730}
AQA M1 2006 January Q2
5 marks Moderate -0.8
2 A particle \(P\) moves with acceleration \(( - 3 \mathbf { i } + 12 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). Initially the velocity of \(P\) is \(4 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the velocity of \(P\) at time \(t\) seconds.
  2. Find the speed of \(P\) when \(t = 0.5\).