Direct collision, find impulse magnitude

Two particles collide directly; given masses and velocities before and after collision, find the magnitude of the impulse exerted by one particle on the other.

32 questions · Moderate -0.6

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Edexcel M1 2010 January Q1
6 marks Moderate -0.8
  1. A particle \(A\) of mass 2 kg is moving along a straight horizontal line with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Another particle \(B\) of mass \(m \mathrm {~kg}\) is moving along the same straight line, in the opposite direction to \(A\), with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The particles collide. The direction of motion of \(A\) is unchanged by the collision. Immediately after the collision, \(A\) is moving with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) is moving with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
    1. the magnitude of the impulse exerted by \(B\) on \(A\) in the collision,
    2. the value of \(m\).
    3. An athlete runs along a straight road. She starts from rest and moves with constant acceleration for 5 seconds, reaching a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). This speed is then maintained for \(T\) seconds. She then decelerates at a constant rate until she stops. She has run a total of 500 m in 75 s .
    4. In the space below, sketch a speed-time graph to illustrate the motion of the athlete.
    5. Calculate the value of \(T\).
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{330c2068-fe0a-4c6d-b892-79ab173c6a11-04_271_750_214_598} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} A particle of mass \(m \mathrm {~kg}\) is attached at \(C\) to two light inextensible strings \(A C\) and \(B C\). The other ends of the strings are attached to fixed points \(A\) and \(B\) on a horizontal ceiling. The particle hangs in equilibrium with \(A C\) and \(B C\) inclined to the horizontal at \(30 ^ { \circ }\) and \(60 ^ { \circ }\) respectively, as shown in Figure 1. Given that the tension in \(A C\) is 20 N , find
  2. the tension in \(B C\),
  3. the value of \(m\).
Edexcel M1 2011 January Q1
5 marks Moderate -0.8
  1. Two particles \(B\) and \(C\) have mass \(m \mathrm {~kg}\) and 3 kg respectively. They are moving towards each other in opposite directions on a smooth horizontal table. The two particles collide directly. Immediately before the collision, the speed of \(B\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(C\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). In the collision the direction of motion of \(C\) is reversed and the direction of motion of \(B\) is unchanged. Immediately after the collision, the speed of \(B\) is \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(C\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Find
  1. the value of \(m\),
  2. the magnitude of the impulse received by \(C\).
Edexcel M1 2012 January Q1
5 marks Moderate -0.8
  1. A railway truck \(P\), of mass \(m \mathrm {~kg}\), is moving along a straight horizontal track with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Truck \(P\) collides with a truck \(Q\) of mass 3000 kg , which is at rest on the same track. Immediately after the collision the speed of \(P\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(Q\) is \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The direction of motion of \(P\) is reversed by the collision.
Modelling the trucks as particles, find
  1. the magnitude of the impulse exerted by \(P\) on \(Q\),
  2. the value of \(m\).
Edexcel M1 2001 June Q1
6 marks Moderate -0.8
  1. Two small balls \(A\) and \(B\) have masses 0.5 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 \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The speed of \(A\) immediately after the collision is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The direction of the motion of \(A\) is unchanged as a result of the collision.
By modelling the balls as particles, find
  1. the speed of \(B\) immediately after the collision,
  2. the magnitude of the impulse exerted on \(B\) in the collision.
Edexcel M1 2006 June Q2
7 marks Moderate -0.8
2. Two particles \(A\) and \(B\) have mass 0.4 kg and 0.3 kg respectively. They are moving in opposite directions on a smooth horizontal table and collide directly. Immediately before the collision, the speed of \(A\) is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). As a result of the collision, the direction of motion of \(B\) is reversed and its speed immediately after the collision is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
  1. the speed of \(A\) immediately after the collision, stating clearly whether the direction of motion of \(A\) is changed by the collision,
  2. the magnitude of the impulse exerted on \(B\) in the collision, stating clearly the units in which your answer is given.
Edexcel M1 2007 June Q2
7 marks Moderate -0.3
2. Two particles \(A\) and \(B\), of mass 0.3 kg and \(m \mathrm {~kg}\) respectively, are moving in opposite directions along the same straight horizontal line so that the particles collide directly. Immediately before the collision, the speeds of \(A\) and \(B\) are \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. In the collision the direction of motion of each particle is reversed and, immediately after the collision, the speed of each particle is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
  1. the magnitude of the impulse exerted by \(B\) on \(A\) in the collision,
  2. the value of \(m\).
Edexcel M1 2009 June Q3
6 marks Moderate -0.3
3. Two particles \(A\) and \(B\) are moving on a smooth horizontal plane. The mass of \(A\) is \(2 m\) and the mass of \(B\) is \(m\). The particles are moving along the same straight line but in opposite directions and they collide directly. Immediately before they collide the speed of \(A\) is \(2 u\) and the speed of \(B\) is \(3 u\). The magnitude of the impulse received by each particle in the collision is \(\frac { 7 m u } { 2 }\). Find
  1. the speed of \(A\) immediately after the collision,
  2. the speed of \(B\) immediately after the collision.
Edexcel M1 2010 June Q1
5 marks Moderate -0.8
  1. A particle \(P\) is moving with constant velocity \(( - 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). At time \(t = 6 \mathrm {~s} P\) is at the point with position vector \(( - 4 \mathbf { i } - 7 \mathbf { j } ) \mathrm { m }\). Find the distance of \(P\) from the origin at time \(t = 2 \mathrm {~s}\).
    (5)
  2. Particle \(P\) has mass \(m \mathrm {~kg}\) and particle \(Q\) has mass \(3 m \mathrm {~kg}\). The particles are moving in opposite directions along a smooth horizontal plane when they collide directly. Immediately before the collision \(P\) has speed \(4 u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(Q\) has speed \(k u \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(k\) is a constant. As a result of the collision the direction of motion of each particle is reversed and the speed of each particle is halved.
    1. Find the value of \(k\).
    2. Find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted on \(P\) by \(Q\).
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{25300ba0-1e54-4242-8db4-a593f5d5a80e-04_195_579_260_507} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} A small box is pushed along a floor. The floor is modelled as a rough horizontal plane and the box is modelled as a particle. The coefficient of friction between the box and the floor is \(\frac { 1 } { 2 }\). The box is pushed by a force of magnitude 100 N which acts at an angle of \(30 ^ { \circ }\) with the floor, as shown in Figure 1. Given that the box moves with constant speed, find the mass of the box.
Edexcel M1 2012 June Q1
6 marks Moderate -0.8
  1. Two particles \(A\) and \(B\), of mass \(5 m \mathrm {~kg}\) and \(2 m \mathrm {~kg}\) respectively, are moving in opposite directions along the same straight horizontal line. The particles collide directly. Immediately before the collision, the speeds of \(A\) and \(B\) are \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The direction of motion of \(A\) is unchanged by the collision. Immediately after the collision, the speed of \(A\) is \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    1. Find the speed of \(B\) immediately after the collision.
    In the collision, the magnitude of the impulse exerted on \(A\) by \(B\) is 3.3 N s .
  2. Find the value of \(m\).
Edexcel M1 2013 June Q1
6 marks Moderate -0.8
  1. Two particles \(A\) and \(B\), of mass 2 kg and 3 kg respectively, are moving towards each other in opposite directions along the same straight line on a smooth horizontal surface. The particles collide directly. Immediately before the collision the speed of \(A\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The magnitude of the impulse exerted on \(B\) by \(A\) is 14 N s . Find
    1. the speed of \(A\) immediately after the collision,
    2. the speed of \(B\) immediately after the collision.
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{278c8424-38aa-48c2-bc82-af4be9234f71-03_359_1298_219_413} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} A particle of weight 8 N is attached at \(C\) to the ends of two light inextensible strings \(A C\) and \(B C\). The other ends, \(A\) and \(B\), are attached to a fixed horizontal ceiling. The particle hangs at rest in equilibrium, with the strings in a vertical plane. The string \(A C\) is inclined at \(35 ^ { \circ }\) to the horizontal and the string \(B C\) is inclined at \(25 ^ { \circ }\) to the horizontal, as shown in Figure 1. Find
    (i) the tension in the string \(A C\),
    (ii) the tension in the string \(B C\).
Edexcel M1 2013 June Q1
6 marks Moderate -0.8
\begin{enumerate} \item Particle \(P\) has mass 3 kg and particle \(Q\) has mass \(m \mathrm {~kg}\). The particles are moving in opposite directions along a smooth horizontal plane when they collide directly. Immediately before the collision, the speed of \(P\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(Q\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). In the collision the direction of motion of \(P\) is unchanged and the direction of motion of \(Q\) is reversed. Immediately after the collision, the speed of \(P\) is \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(Q\) is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the magnitude of the impulse exerted on \(P\) in the collision.
  2. Find the value of \(m\).
    \item A woman travels in a lift. The mass of the woman is 50 kg and the mass of the lift is 950 kg . The lift is being raised vertically by a vertical cable which is attached to the top of the lift. The lift is moving upwards and has constant deceleration of \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). By modelling the cable as being light and inextensible, find
Edexcel M1 2015 June Q1
6 marks Moderate -0.5
  1. Particle \(P\) of mass \(m\) and particle \(Q\) of mass \(k m\) are moving in opposite directions on a smooth horizontal plane when they collide directly. Immediately before the collision the speed of \(P\) is \(5 u\) and the speed of \(Q\) is \(u\). Immediately after the collision the speed of each particle is halved and the direction of motion of each particle is reversed.
Find
  1. the value of \(k\),
  2. the magnitude of the impulse exerted on \(P\) by \(Q\) in the collision.
Edexcel M1 2017 June Q2
7 marks Moderate -0.8
2. Two particles, \(P\) and \(Q\), have masses \(2 m\) and \(3 m\) respectively. They are moving towards each other in opposite directions on a smooth horizontal plane when they collide directly. Immediately before they collide the speed of \(P\) is \(4 u\) and the speed of \(Q\) is \(3 u\). As a result of the collision, \(Q\) has its direction of motion reversed and is moving with speed \(u\).
  1. Find the speed of \(P\) immediately after the collision.
  2. State whether or not the direction of motion of \(P\) has been reversed by the collision.
  3. Find the magnitude of the impulse exerted on \(P\) by \(Q\) in the collision.
Edexcel M1 2018 June Q1
6 marks Moderate -0.8
  1. Two particles, \(P\) and \(Q\), have masses \(3 m\) and \(m\) respectively. They are moving in opposite directions towards each other along the same straight line on a smooth horizontal plane and collide directly. The speeds of \(P\) and \(Q\) immediately before the collision are \(2 u\) and \(4 u\) respectively. The magnitude of the impulse received by each particle in the collision is \(\frac { 21 m u } { 4 }\).
    1. Find the speed of \(P\) after the collision.
    2. Find the speed of \(Q\) after the collision.
Edexcel M1 2016 January Q2
8 marks Moderate -0.5
  1. Two particles \(P\) and \(Q\) are moving in opposite directions along the same horizontal straight line. Particle \(P\) is moving due east and particle \(Q\) is moving due west. Particle \(P\) has mass \(2 m\) and particle \(Q\) has mass \(3 m\). The particles collide directly. Immediately before the collision, the speed of \(P\) is \(4 u\) and the speed of \(Q\) is \(u\). The magnitude of the impulse in the collision is \(\frac { 33 } { 5 } m u\).
    1. Find the speed and direction of motion of \(P\) immediately after the collision.
    2. Find the speed and direction of motion of \(Q\) immediately after the collision.
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    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{054e11cb-9416-40c0-9dde-8d12818bab3f-04_268_862_123_543} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} A boy is pulling a sledge of mass 8 kg in a straight line at a constant speed across rough horizontal ground by means of a rope. The rope is inclined at \(30 ^ { \circ }\) to the ground, as shown in Figure 1. The coefficient of friction between the sledge and the ground is \(\frac { 1 } { 5 }\). By modelling the sledge as a particle and the rope as a light inextensible string, find the tension in the rope.
Edexcel M1 2019 January Q1
6 marks Moderate -0.8
  1. Two particles, \(A\) and \(B\), have masses \(2 m\) and \(3 m\) respectively. They are moving towards each other in opposite directions along the same straight line on a smooth horizontal plane when they collide directly. Immediately before they collide, the speed of \(A\) is \(3 u\) and the speed of \(B\) is \(u\). As a result of the collision, the speed of \(A\) is halved and the direction of motion of each particle is reversed.
    1. Find the speed of \(B\) immediately after the collision.
    2. Find the magnitude of the impulse exerted on \(A\) by \(B\) in the collision.
Edexcel M1 2021 January Q2
6 marks Moderate -0.3
2. Two particles, \(P\) and \(Q\), have masses \(2 m\) and \(m\) respectively. The particles are moving towards each other in opposite directions along the same straight line on a smooth horizontal plane. The particles collide directly. Immediately before the collision, the speed of \(P\) is \(3 u\) and the speed of \(Q\) is \(2 u\). The magnitude of the impulse exerted on \(Q\) by \(P\) in the collision is 5mu. Find
  1. the speed of \(P\) immediately after the collision,
  2. the speed of \(Q\) immediately after the collision.
Edexcel M1 2022 January Q2
7 marks Moderate -0.3
2. A particle \(P\) has mass \(k m\) and a particle \(Q\) has mass \(m\). The particles are moving towards each other in opposite directions along the same straight line when they collide directly. Immediately before the collision, \(P\) has speed \(3 u\) and \(Q\) has speed \(u\).
As a result of the collision, the direction of motion of each particle is reversed and the speed of each particle is halved.
  1. Find the value of \(k\).
  2. Find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted on \(Q\) in the collision.
Edexcel M1 2018 June Q1
6 marks Moderate -0.8
  1. Particle \(P\) has mass \(3 m\) and particle \(Q\) has mass \(m\). The particles are moving towards each other in opposite directions along the same straight line on a smooth horizontal plane. The particles collide directly. Immediately before the collision the speed of \(P\) is \(u\) and the speed of \(Q\) is \(3 u\). In the collision, the magnitude of the impulse exerted by \(Q\) on \(P\) is \(5 m u\).
    1. Find the speed of \(P\) immediately after the collision.
    2. Find the speed of \(Q\) immediately after the collision.
Edexcel M1 2022 June Q1
4 marks Moderate -0.8
  1. Two particles, \(P\) and \(Q\), are moving towards each other in opposite directions along the same straight line when they collide directly. Immediately before the collision the speed of \(Q\) is \(2 u\). The mass of \(Q\) is \(3 m\) and the magnitude of the impulse exerted by \(P\) on \(Q\) in the collision is \(4 m u\).
Find
  1. the speed of \(Q\) immediately after the collision,
  2. the direction of motion of \(Q\) immediately after the collision.
Edexcel M1 2023 June Q1
7 marks Moderate -0.8
  1. A particle \(A\) has mass 4 kg and a particle \(B\) has mass 2 kg .
The particles move towards each other in opposite directions along the same straight line on a smooth horizontal table and collide directly. Immediately before the collision, the speed of \(A\) is \(2 u \mathrm {~ms} ^ { - 1 }\) and the speed of \(B\) is \(3 u \mathrm {~ms} ^ { - 1 }\) Immediately after the collision, the speed of \(B\) is \(2 u \mathrm {~ms} ^ { - 1 }\) The direction of motion of \(B\) is reversed by the collision.
  1. Find, in terms of \(u\), the speed of \(A\) immediately after the collision.
  2. State the direction of motion of \(A\) immediately after the collision.
  3. Find, in terms of \(u\), the magnitude of the impulse received by \(B\) in the collision. State the units of your answer. \section*{[In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal perpendicular unit vectors.]}
Edexcel M1 2016 October Q1
7 marks Moderate -0.8
  1. Two particles, \(P\) and \(Q\), have masses \(2 m\) and \(3 m\) respectively. They are moving towards each other, in opposite directions, along the same straight line, on a smooth horizontal plane. The particles collide. Immediately before they collide the speed of \(P\) is \(2 u\) and the speed of \(Q\) is \(u\). In the collision the magnitude of the impulse exerted on \(P\) by \(Q\) is \(5 m u\).
    1. Find the speed of \(P\) immediately after the collision.
    2. State whether the direction of motion of \(P\) has been reversed by the collision.
    3. Find the speed of \(Q\) immediately after the collision.
Edexcel M1 2018 October Q1
6 marks Moderate -0.8
  1. A particle \(P\) of mass 0.8 kg is moving along a straight horizontal line on a smooth hoizontal surface with speed \(4 \mathrm {~ms} ^ { - 1 }\). A second particle \(Q\) of mass 2 kg is moving, in the opposite direction to \(P\), along the same straight line with speed \(2 \mathrm {~ms} ^ { - 1 }\). The particles collide directly. Immediately after the collision the direction of motion of each particle is reversed and the speed of \(P\) is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    1. Find the speed of \(Q\) immediately after the collision.
    2. Find the magnitude of the impulse exerted by \(Q\) on \(P\) in the collision, stating the units of your answer.
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    Figure 1 A non-uniform plank \(A B\) has weight 60 N and length 5 m . The plank rests horizontally in equilibrium on two smooth supports at \(A\) and \(C\), where \(A C = 3 \mathrm {~m}\), as shown in Figure 1. A parcel of weight 12 N is placed on the plank at \(B\) and the plank remains horizontal and in equilibrium. The magnitude of the reaction of the support at \(A\) on the plank is half the magnitude of the reaction of the support at \(C\) on the plank. By modelling the plank as a non-uniform rod and the parcel as a particle,
  2. find the distance of the centre of mass of the plank from \(A\).
  3. State briefly how you have used the modelling assumption
    1. that the parcel is a particle,
    2. that the plank is a rod.
Edexcel M1 2018 Specimen Q2
6 marks Moderate -0.3
2. Two particles \(P\) and \(Q\) are moving in opposite directions along the same horizontal straight line. Particle \(P\) has mass \(m\) and particle \(Q\) has mass \(k m\). The particles collide directly. Immediately before the collision, the speed of \(P\) is \(u\) and the speed of \(Q\) is \(2 u\). As a result of the collision, the direction of motion of each particle is reversed and the speed of each particle is halved.
  1. Find the value of \(k\).
  2. Find, in terms of \(m\) and \(u\) only, the magnitude of the impulse exerted on \(Q\) by \(P\) in the collision.
Edexcel M1 2005 January Q1
7 marks Moderate -0.8
  1. A particle \(P\) of mass 1.5 kg is moving along a straight horizontal line with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Another particle \(Q\) of mass 2.5 kg is moving, in the opposite direction, along the same straight line with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The particles collide. Immediately after the collision the direction of motion of \(P\) is reversed and its speed is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    1. Calculate the speed of \(Q\) immediately after the impact.
    2. State whether or not the direction of motion of \(Q\) is changed by the collision.
    3. Calculate the magnitude of the impulse exerted by \(Q\) on \(P\), giving the units of your answer.
    \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{975a5462-0ba3-448a-8a5a-fc90da640d57-03_521_858_306_552}
    \end{figure} A plank \(A B\) has mass 40 kg and length 3 m . A load of mass 20 kg is attached to the plank at \(B\). The loaded plank is held in equilibrium, with \(A B\) horizontal, by two vertical ropes attached at \(A\) and \(C\), as shown in Figure 1. The plank is modelled as a uniform rod and the load as a particle. Given that the tension in the rope at \(C\) is three times the tension in the rope at \(A\), calculate
  2. the tension in the rope at \(C\),
  3. the distance \(C B\).