Direct collision, find mass

Two particles collide directly; given initial and final velocities and one mass, find the unknown mass using conservation of momentum.

11 questions · Moderate -0.5

6.03b Conservation of momentum: 1D two particles
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CAIE M1 2020 June Q1
3 marks Easy -1.2
1 Particles \(P\) of mass \(m \mathrm {~kg}\) and \(Q\) of mass 0.2 kg are free to move on a smooth horizontal plane. \(P\) is projected at a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) towards \(Q\) which is stationary. After the collision \(P\) and \(Q\) move in opposite directions with speeds of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. Find \(m\).
Edexcel M1 2024 January Q2
6 marks Moderate -0.8
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e59a66b8-c2ad-41fd-9959-9d21e9455c37-04_204_947_242_559} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows two particles, \(A\) and \(B\), moving in opposite directions on a smooth horizontal surface. Particle \(A\) has mass 5 kg and particle \(B\) has mass \(x \mathrm {~kg}\). The particles collide directly.
Immediately before the collision, the speed of \(A\) is \(3 \mathrm {~ms} ^ { - 1 }\) and the speed of \(B\) is \(x \mathrm {~ms} ^ { - 1 }\) Immediately after the collision, the speed of \(A\) is \(1 \mathrm {~ms} ^ { - 1 }\) and its direction of motion is unchanged. Immediately after the collision, the speed of \(B\) is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
  1. Find the value of \(x\).
  2. Find the magnitude of the impulse exerted on \(A\) by \(B\) in the collision.
Edexcel M1 2014 June Q1
5 marks Moderate -0.8
  1. Two small smooth balls \(A\) and \(B\) have mass 0.6 kg and 0.9 kg respectively. They are moving in a straight line towards each other in opposite directions on a smooth horizontal floor and collide directly. Immediately before the collision the speed of \(A\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The speed of \(A\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) immediately after the collision and \(B\) is brought to rest by the collision.
Find
  1. the value of \(v\),
  2. the magnitude of the impulse exerted on \(A\) by \(B\) in the collision.
Edexcel M1 2008 January Q1
6 marks Moderate -0.8
  1. Two particles \(A\) and \(B\) have masses 4 kg and \(m \mathrm {~kg}\) 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 \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after the collision, the direction of motion of \(A\) is unchanged and the speed of \(A\) is \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    1. Find the magnitude of the impulse exerted on \(A\) in the collision.
    Immediately after the collision, the speed of \(B\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the value of \(m\).
OCR M1 2006 June Q1
5 marks Moderate -0.8
1 Each of two wagons has an unloaded mass of 1200 kg . One of the wagons carries a load of mass \(m \mathrm {~kg}\) and the other wagon is unloaded. The wagons are moving towards each other on the same rails, each with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when they collide. Immediately after the collision the loaded wagon is at rest and the speed of the unloaded wagon is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the value of \(m\).
Edexcel M1 2016 June Q2
6 marks Moderate -0.3
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 \(km\). The particles collide directly. Immediately before the collision, the speed of \(P\) is \(u\) and the speed of \(Q\) is \(2u\). 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 \(k\). [4]
  2. Find, in terms of \(m\) and \(u\) only, the magnitude of the impulse exerted on \(Q\) by \(P\) in the collision. [2]
Edexcel M1 Specimen Q2
7 marks Moderate -0.3
Particle \(P\) has mass \(m\) kg and particle \(Q\) has mass \(3m\) kg. The particles are moving in opposite directions along a smooth horizontal plane when they collide directly. Immediately before the collision \(P\) has speed \(4u\) m s\(^{-1}\) and \(Q\) has speed \(ku\) m 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 \(k\). [4]
  2. Find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted on \(P\) by \(Q\). [3]
Edexcel M1 2004 January Q1
7 marks Moderate -0.8
Two trucks \(A\) and \(B\), moving in opposite directions on the same horizontal railway track, collide. The mass of \(A\) is 600 kg. The mass of \(B\) is \(m\) kg. Immediately before the collision, the speed of \(A\) is 4 m s\(^{-1}\) and the speed of \(B\) is 2 m s\(^{-1}\). Immediately after the collision, the trucks are joined together and move with the same speed 0.5 m s\(^{-1}\). The direction of motion of \(A\) is unchanged by the collision. Find
  1. the value of \(m\), [4]
  2. the magnitude of the impulse exerted on \(A\) in the collision. [3]
OCR M1 Q1
5 marks Moderate -0.3
Each of two wagons has an unloaded mass of \(1200\) kg. One of the wagons carries a load of mass \(m\) kg and the other wagon is unloaded. The wagons are moving towards each other on the same rails, each with speed \(3\) m s\(^{-1}\), when they collide. Immediately after the collision the loaded wagon is at rest and the speed of the unloaded wagon is \(5\) m s\(^{-1}\). Find the value of \(m\). [5]
OCR M1 2009 June Q5
11 marks Standard +0.3
  1. \includegraphics{figure_5_1} A particle \(P\) of mass \(0.5\) kg is projected with speed \(6\) m s\(^{-1}\) on a smooth horizontal surface towards a stationary particle \(Q\) of mass \(m\) kg (see Fig. 1). After the particles collide, \(P\) has speed \(v\) m s\(^{-1}\) in the original direction of motion, and \(Q\) has speed \(1\) m s\(^{-1}\) more than \(P\). Show that \(v(m + 0.5) = -m + 3\). [3]
  2. \includegraphics{figure_5_2} \(Q\) and \(P\) are now projected towards each other with speeds \(4\) m s\(^{-1}\) and \(2\) m s\(^{-1}\) respectively (see Fig. 2). Immediately after the collision the speed of \(Q\) is \(v\) m s\(^{-1}\) with its direction of motion unchanged and \(P\) has speed \(1\) m s\(^{-1}\) more than \(Q\). Find another relationship between \(m\) and \(v\) in the form \(v(m + 0.5) = am + b\), where \(a\) and \(b\) are constants. [4]
  3. By solving these two simultaneous equations show that \(m = 0.9\), and hence find \(v\). [4]
Edexcel M1 Q1
7 marks Standard +0.3
Two particles \(P\) and \(Q\), of mass \(m\) and \(km\) respectively, are travelling in opposite directions on a straight horizontal path with speeds \(3u\) and \(2u\) respectively. \(P\) and \(Q\) collide and, as a result, the direction of motion of both particles is reversed and their speeds are halved.
  1. Find the value of \(k\). [4 marks]
  2. Write down an expression in terms of \(m\) and \(u\) for the magnitude of the impulse which \(P\) exerts on \(Q\) during the collision. [3 marks]