Coalescence collision

2 Two small smooth spheres \(A\) and \(B\), of equal radii and of masses km kg and \(m \mathrm {~kg}\) respectively, where \(k > 1\), are free to move on a smooth horizontal plane. \(A\) is moving towards \(B\) with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) is moving towards \(A\) with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After the collision \(A\) and \(B\) coalesce and move with speed \(4 \mathrm {~ms} ^ { - 1 }\).

1. Find \(k\).
2. Find, in terms of \(m\), the loss of kinetic energy due to the collision.

2 An ice skater of mass 40 kg is moving in a straight line with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when she collides with a skater of mass 60 kg moving in the opposite direction along the same straight line with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After the collision the skaters move together with a common speed in the same straight line. Calculate their common speed, and state their direction of motion.

7 \includegraphics[max width=\textwidth, alt={}, center]{470e70de-66ba-4dcc-a205-0c92f29471b1-4_227_901_1352_623} Two particles \(P\) and \(Q\) have masses 0.7 kg and 0.3 kg respectively. \(P\) and \(Q\) are simultaneously projected towards each other in the same straight line on a horizontal surface with initial speeds of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively (see diagram). Before \(P\) and \(Q\) collide the only horizontal force acting on each particle is friction and each particle decelerates at \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The particles coalesce when they collide.

1. Given that \(P\) and \(Q\) collide 2 s after projection, calculate the speed of each particle immediately before the collision, and the speed of the combined particle immediately after the collision.
2. Given instead that \(P\) and \(Q\) collide 3 s after projection,
   1. sketch on a single diagram the \(( t , v )\) graphs for the two particles in the interval \(0 \leqslant t < 3\),
   2. calculate the distance between the two particles at the instant when they are projected.

6 Particle \(P\) of mass 0.3 kg and particle \(Q\) of mass 0.2 kg are 3.6 m apart on a smooth horizontal surface. \(P\) and \(Q\) are simultaneously projected directly towards each other along a straight line. Before the particles collide \(P\) has speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(Q\) has speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Given that the particles coalesce in the collision, calculate their common speed after they collide.
2. It is given instead that one particle is at rest immediately after the collision.
   1. State which particle is in motion after the collision and find the speed of this particle.
   2. Find the time taken after the collision for the moving particle to return to its initial position.
   3. On a single diagram sketch the \(( t , v )\) graphs for the two particles, with \(t = 0\) as the instant of their initial projection. \(7 \quad A\) and \(B\) are two points on a line of greatest slope of a plane inclined at \(45 ^ { \circ }\) to the horizontal and \(A B = 2 \mathrm {~m}\). A particle \(P\) of mass 0.4 kg is projected from \(A\) towards \(B\) with speed \(5 \mathrm {~ms} ^ { - 1 }\). The coefficient of friction between the plane and \(P\) is 0.2 .
      1. Given that the level of \(A\) is above the level of \(B\), calculate the speed of \(P\) when it passes through the point \(B\), and the time taken to travel from \(A\) to \(B\).
      2. Given instead that the level of \(A\) is below the level of \(B\),
         (a) show that \(P\) does not reach \(B\),
         (b) calculate the difference in the momentum of \(P\) for the two occasions when it is at \(A\).

7 \includegraphics[max width=\textwidth, alt={}, center]{f0813713-d677-4ed7-87e1-971a64bdb6ff-4_122_255_1503_561} The diagram shows two particles \(P\) and \(Q\), of masses 0.2 kg and 0.3 kg respectively, which move on a horizontal surface in the same direction along a straight line. A stationary particle \(R\) of mass 1.5 kg also lies on this line. \(P\) and \(Q\) collide and coalesce to form a combined particle \(C\). Immediately before this collision \(P\) has velocity \(4 \mathrm {~ms} ^ { - 1 }\) and \(Q\) has velocity \(2.5 \mathrm {~ms} ^ { - 1 }\).

1. Calculate the velocity of \(C\) immediately after this collision. At time \(t \mathrm {~s}\) after this collision the velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of \(C\) is given by \(v = V _ { 0 } - 3 t ^ { 2 }\) for \(0 < t \leqslant 0.3\). \(C\) strikes \(R\) when \(t = 0.3\).
2. (a) State the value of \(V _ { 0 }\).
   (b) Calculate the distance \(C\) moves before it strikes \(R\).
   (c) Find the acceleration of \(C\) immediately before it strikes \(R\). Immediately after \(C\) strikes \(R\), the particles have equal speeds but move in opposite directions.
3. Find the speed of \(C\) immediately after it strikes \(R\).

1 Two particles \(A\) and \(B\) have masses of 3 kg and 2 kg respectively. They are moving along a straight horizontal line towards each other. Each particle is moving with a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when they collide. \includegraphics[max width=\textwidth, alt={}, center]{965a176a-848c-478d-a748-80fc9dfe4399-2_225_579_676_660}

1. If the particles coalesce during the collision to form a single particle, find the speed of the combined particle after the collision.
2. If, after the collision, \(A\) moves in the same direction as before the collision with speed \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the speed of \(B\) after the collision.

1 Two particles, \(A\) and \(B\), are travelling in the same direction with constant speeds along a straight line when they collide. Particle \(A\) has mass 2.5 kg and speed \(12 \mathrm {~ms} ^ { - 1 }\). Particle \(B\) has mass 1.5 kg and speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After the collision, the two particles move together at the same speed. Find the speed of the particles after the collision.

1 A trolley, of mass 5 kg , is moving in a straight line on a smooth horizontal surface. It has a velocity of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it collides with a stationary trolley, of mass \(m \mathrm {~kg}\). Immediately after the collision, the trolleys move together with velocity \(2.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find \(m\).
(3 marks)

2 Two toy trains, \(A\) and \(B\), are moving in the same direction on a straight horizontal track when they collide. As they collide, the speed of \(A\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after the collision, they move together with a speed of \(3.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The mass of \(A\) is 2 kg . Find the mass of \(B\).

1 A toy train of mass 300 grams is moving along a straight horizontal track at a speed of \(2.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). This toy train collides with another toy train, of mass 200 grams, which is at rest on the same track. During the collision, the two trains lock together and then move together. Find the speed of the trains immediately after the collision.

1 A child, of mass 48 kg , is initially standing at rest on a stationary skateboard. The child jumps off the skateboard and initially moves horizontally with a speed of \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The skateboard moves with a speed of \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the opposite direction to the direction of motion of the child. Find the mass of the skateboard.
[0pt] [3 marks]

5. A particle \(A\), of mass \(m \mathrm {~kg}\), has position vector \(11 \mathbf { i } + 6 \mathbf { j }\) and a velocity \(2 \mathbf { i } + 7 \mathbf { j }\). At the same moment, second particle \(B\), of mass \(2 m \mathrm {~kg}\), has position vector \(7 \mathbf { i } + 10 \mathbf { j }\) and a velocity \(5 \mathbf { i } + 4 \mathbf { j }\).

1. If the particles continue to move with these velocities, prove that the particles will collide. Given that the particles coalesce after collision, find the common velocity of the particles after collision.
2. Determine the impulse exerted by \(A\) on \(B\).
3. Calculate the loss of kinetic energy caused by the collision.

1 Two particles, \(A\) and \(B\), are travelling in the same direction along a straight line on a smooth horizontal surface. Particle \(A\) has mass 3 kg and particle \(B\) has mass 7 kg . Particle \(A\) has a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and particle \(B\) has a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{fe8c1ea4-cf4d-4741-8af5-03e8c2c88559-2_186_835_653_593} Particle \(A\) and particle \(B\) collide and coalesce to form a single particle. Find the speed of this single particle after the collision.

2 Two particles, \(A\) and \(B\), are moving on a smooth horizontal surface. Particle \(A\) has mass 2 kg and velocity \(\left[ \begin{array} { r } 3 \\ - 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\). Particle \(B\) has mass 3 kg and velocity \(\left[ \begin{array} { r } - 4 \\ 1 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\). The two particles collide, and they coalesce during the collision.

1. Find the velocity of the combined particles after the collision.
2. Find the speed of the combined particles after the collision.

1 A particle \(A\), of mass 0.2 kg , collides with a particle \(B\), of mass 0.3 kg Immediately before the collision, the velocity of \(A\) is \(\left[ \begin{array} { c } 4 \\ 12 \end{array} \right] \mathrm { ms } ^ { - 1 }\) and the velocity of \(B\) is \(\left[ \begin{array} { l } - 1 \\ - 3 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) As a result of the collision the particles coalesce to become a single particle.
Find the velocity of the single particle.
Circle your answer.
[0pt] [1 mark] $$\left[ \begin{array} { l } 0.5 \\ 1.5 \end{array} \right] \mathrm { m } \mathrm {~s} ^ { - 1 } \quad \left[ \begin{array} { l } 2 \\ 6 \end{array} \right] \mathrm { m } \mathrm {~s} ^ { - 1 } \quad \left[ \begin{array} { l } 1 \\ 3 \end{array} \right] \mathrm { ms } ^ { - 1 } \quad \left[ \begin{array} { l } 3 \\ 9 \end{array} \right] \mathrm { m } \mathrm {~s} ^ { - 1 }$$

A particle \(B\) of mass 5 kg is at rest on a smooth horizontal table. A particle \(A\) of mass 2.5 kg moves on the table with a speed of \(6 \text{ m s}^{-1}\) and collides directly with \(B\). In the collision the two particles coalesce.

1. Find the speed of the combined particle after the collision. [2]
2. Find the loss of kinetic energy of the system due to the collision. [3]

A railway truck \(A\) of mass \(m\) and a second railway truck \(B\) of mass \(4m\) are moving in opposite directions on a smooth straight horizontal track when they collide directly. Immediately before the collision the speed of truck \(A\) is \(3u\) and the speed of truck \(B\) is \(2u\). In the collision the trucks join together. Modelling the trucks as particles, find

1. the speed of \(A\) immediately after the collision, [3]
2. the direction of motion of \(A\) immediately after the collision, [1]
3. the magnitude of the impulse exerted by \(A\) on \(B\) in the collision. [3]

Two particles \(P\) and \(Q\) have masses \(4m\) 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. Immediately before the collision the speed of \(P\) is \(2u\) and the speed of \(Q\) is \(4u\). In the collision, the particles join together to form a single particle. Find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted by \(P\) on \(Q\) in the collision. [6]

A railway truck \(S\) of mass 20 tonnes is moving along a straight horizontal track when it collides with another railway truck \(T\) of mass 30 tonnes which is at rest. Immediately before the collision the speed of \(S\) is \(4\text{ ms}^{-1}\) As a result of the collision, the two railway trucks join together. Find

1. the common speed of the railway trucks immediately after the collision, [2]
2. the magnitude of the impulse exerted on \(S\) in the collision, stating the units of your answer. [3]

A railway truck \(A\) of mass 1800 kg is moving along a straight horizontal track with speed 4 m s\(^{-1}\). It collides directly with a stationary truck \(B\) of mass 1200 kg on the same track. In the collision, \(A\) and \(B\) are coupled and move off together.

1. Find the speed of the trucks immediately after the collision. [3]

After the collision, the trucks experience a constant resistive force of magnitude \(R\) newtons. They come to rest 8 s after the collision.

1. Find \(R\). [3]

A truck of mass 3 tonnes moves on straight horizontal rails. It collides with truck \(B\) of mass 1 tonne, which is moving on the same rails. Immediately before the collision, the speed of \(A\) is \(3 \text{ m s}^{-1}\), the speed of \(B\) is \(4 \text{ m s}^{-1}\), and the trucks are moving towards each other. In the collision, the trucks couple to form a single body \(C\), which continues to move on the rails.

1. Find the speed and direction of \(C\) after the collision. [4]
2. Find, in Ns, the magnitude of the impulse exerted by \(B\) on \(A\) in the collision. [3]
3. State a modelling assumption which you have made about the trucks in your solution [1]

Immediately after the collision, a constant braking force of magnitude 250 N is applied to \(C\). It comes to rest in a distance \(d\) metres.

1. Find the value of \(d\). [4]

Two railway trucks \(A\) and \(B\), of masses 10 000 kg and 7 000 kg respectively, are moving towards each other along a horizontal straight track. The trucks collide, and in the collision \(A\) exerts an impulse on \(B\) of magnitude 84 000 Ns. Immediately after the collision, the trucks move together with speed 10 ms\(^{-1}\). Modelling the trucks as particles,

1. find the speed of each truck immediately before the collision. [6 marks]

When the trucks are moving together along the track, the coefficient of friction between them and the track is 0.15. Assuming that no other resisting forces act on the trucks, calculate

1. the magnitude of the resisting force on the trucks, [3 marks]
2. the time taken after the collision for the trucks to come to rest. [3 marks]

By burning a charge, a cannon fires a cannon ball of mass 12 kg horizontally. As the cannon ball leaves the cannon, its speed is 600 ms\(^{-1}\). The recoiling part of the cannon has a mass of 1600 kg.

1. Determine the speed of the recoiling part immediately after the cannon ball leaves the cannon. [3]
2. Find the energy created by the burning of the charge. State any assumption you have made in your solution and briefly explain how the assumption affects your answer. [5]
3. Calculate the constant force needed to bring the recoiling part to rest in 1.2 m. State, with a reason, whether your answer is an overestimate or an underestimate of the actual force required. [4]