Momentum and Collisions

A ball of mass 0.3 kg is moving vertically downwards with speed 8 m s\(^{-1}\) when it hits the floor which is smooth and horizontal. It rebounds vertically from the floor with speed 6 m s\(^{-1}\). Find the magnitude of the impulse exerted by the floor on the ball. [3]

A ball of mass 0.3 kg is moving vertically downwards with speed 8 m s\(^{-1}\) when it hits the floor which is smooth and horizontal. It rebounds vertically from the floor with speed 6 m s\(^{-1}\). Find the magnitude of the impulse exerted by the floor on the ball. [3]

7. Two particles \(A\) and \(B\), of mass \(3 M \mathrm {~kg}\) and \(2 M \mathrm {~kg}\) respectively, are moving towards each other on a rough horizontal track. Just before they collide, \(A\) has speed \(3 \mathrm {~ms} ^ { - 1 }\) and \(B\) has speed \(5 \mathrm {~ms} ^ { - 1 }\). Immediately after the impact, the direction of motion of both particles has been reversed and they are both travelling at the same speed, \(v\).

1. Show that \(v = 1 \mathrm {~ms} ^ { - 1 }\). The magnitude of the impulse exerted on \(A\) during the collision is 24 Ns.
2. Find the value of \(M\). Given that the coefficient of friction between \(A\) and the track is 0.1 ,
3. find the time taken from the moment of impact until \(A\) comes to rest. END

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)

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] $$\left[ \begin{array} { l } 0.5 \\ 1.5 \end{array} \right] \mathrm { m } \mathrm {~s} ^ { - 1 } \quad \left[ \begin{array} { l } 2 \\ 6 \end{array} \right] \mathrm { m } \mathrm {~s} ^ { - 1 } \quad \left[ \begin{array} { l } 1 \\ 3 \end{array} \right] \mathrm { ms } ^ { - 1 } \quad \left[ \begin{array} { l } 3 \\ 9 \end{array} \right] \mathrm { m } \mathrm {~s} ^ { - 1 }$$

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.

Two particles, \(A\) and \(B\), of masses 3 kg and 6 kg respectively, lie on a smooth horizontal plane. Initially, \(B\) is at rest and \(A\) is moving towards \(B\) with speed 8 ms\(^{-1}\). After \(A\) and \(B\) collide, \(A\) moves with speed 2 ms\(^{-1}\). Find the greater of the two possible total losses of kinetic energy due to the collision. [6]

1. A particle \(P\) has mass \(3 m\) and a particle \(Q\) has mass \(5 m\). The particles 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 \(P\) is \(k u\), where \(k\) is a constant, and the speed of \(Q\) is \(2 u\). Immediately after the collision the speed of \(P\) is \(u\) and the speed of \(Q\) is \(3 u\).
The direction of motion of \(Q\) is reversed by the collision.

1. Find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted on \(Q\) by \(P\) in the collision.
2. Find the two possible values of \(k\). \includegraphics[max width=\textwidth, alt={}, center]{5a2cf693-d966-4787-8778-ecc8a79a6265-03_2647_1837_118_114}

Two particles, \(A\) and \(B\), of masses 3 kg and 6 kg respectively, lie on a smooth horizontal plane. Initially, \(B\) is at rest and \(A\) is moving towards \(B\) with speed 8 ms\(^{-1}\). After \(A\) and \(B\) collide, \(A\) moves with speed 2 ms\(^{-1}\). Find the greater of the two possible total losses of kinetic energy due to the collision. [6]

1 Two particles \(P\) and \(Q\), of masses 0.3 kg and 0.2 kg respectively, are at rest on a smooth horizontal plane. \(P\) is projected at a speed of \(4 \mathrm {~ms} ^ { - 1 }\) directly towards \(Q\). After \(P\) and \(Q\) collide, \(Q\) begins to move with a speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the speed of \(P\) after the collision.
   After the collision, \(Q\) moves directly towards a third particle \(R\), of mass \(m \mathrm {~kg}\), which is at rest on the plane. The two particles \(Q\) and \(R\) coalesce on impact and move with a speed of \(2 \mathrm {~ms} ^ { - 1 }\).
2. Find \(m\).

1 Two particles \(P\) and \(Q\), of masses 0.3 kg and 0.2 kg respectively, are at rest on a smooth horizontal plane. \(P\) is projected at a speed of \(4 \mathrm {~ms} ^ { - 1 }\) directly towards \(Q\). After \(P\) and \(Q\) collide, \(Q\) begins to move with a speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the speed of \(P\) after the collision.
   After the collision, \(Q\) moves directly towards a third particle \(R\), of mass \(m \mathrm {~kg}\), which is at rest on the plane. The two particles \(Q\) and \(R\) coalesce on impact and move with a speed of \(2 \mathrm {~ms} ^ { - 1 }\).
2. Find \(m\).

3 Three small smooth spheres \(A , B\) and \(C\) of equal radii and of masses \(4 \mathrm {~kg} , 2 \mathrm {~kg}\) and 3 kg respectively, lie in that order in a straight line on a smooth horizontal plane. Initially, \(B\) and \(C\) are at rest and \(A\) is moving towards \(B\) with speed \(6 \mathrm {~ms} ^ { - 1 }\). After the collison with \(B\), sphere \(A\) continues to move in the same direction but with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the speed of \(B\) after this collison.
   Sphere \(B\) collides with \(C\). In this collison these two spheres coalesce to form an object \(D\).
2. Find the speed of \(D\) after this collision.
3. Show that the total loss of kinetic energy in the system due to the two collisions is 38.4 J .

A small ball \(A\) is moving with velocity \((7\mathbf{i} + 12\mathbf{j})\) ms\(^{-1}\). It collides in mid-air with another ball \(B\), of mass \(0.4\) kg, moving with velocity \((-\mathbf{i} + 7\mathbf{j})\) ms\(^{-1}\). Immediately after the collision, \(A\) has velocity \((-3\mathbf{i} + 4\mathbf{j})\) ms\(^{-1}\) and \(B\) has velocity \((6.5\mathbf{i} + 13\mathbf{j})\) ms\(^{-1}\). Calculate the mass of \(A\). [4 marks]

2 A particle, \(A\), of mass 12 kg is moving on a smooth horizontal surface with velocity \(\left[ \begin{array} { l } 4 \\ 7 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\). It then collides and coalesces with a second particle, \(B\), of mass 4 kg .

1. If before the collision the velocity of \(B\) was \(\left[ \begin{array} { l } 2 \\ 3 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\), find the velocity of the combined particle after the collision.
2. If after the collision the velocity of the combined particle is \(\left[ \begin{array} { l } 1 \\ 4 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\), find the velocity of \(B\) before the collision.

Two smooth spheres \(A\) and \(B\), of masses \(2m\) and \(3m\) respectively, are moving on a smooth horizontal table with velocities \((3\mathbf{i} - \mathbf{j})\) ms\(^{-1}\) and \((4\mathbf{i} + \mathbf{j})\) ms\(^{-1}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors. They collide, after which \(A\) has velocity \((5\mathbf{i} + \mathbf{j})\) ms\(^{-1}\).

1. Find the magnitude of the impulse exerted on \(B\) by \(A\), stating the units of your answer. [4 marks]
2. Find the speed of \(B\) immediately after the collision. [5 marks]

A small ball of mass \(0.8\) kg is moving with speed \(10.5\) m s\(^{-1}\) when it receives an impulse of magnitude \(4\) N s. The speed of the ball immediately afterwards is \(8.5\) m s\(^{-1}\). The angle between the directions of motion before and after the impulse acts is \(\alpha\). Using an impulse-momentum triangle, or otherwise, find \(\alpha\). [6]

1. A particle \(P\) of mass 0.5 kg is moving with velocity ( \(4 \mathbf { i } + 3 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse \(\mathbf { J }\) Ns. Immediately after receiving the impulse, \(P\) is moving with velocity \(( - \mathbf { i } + 6 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
   1. Find the magnitude of \(\mathbf { J }\).
   The angle between the direction of the impulse and the direction of motion of \(P\) immediately before receiving the impulse is \(\alpha ^ { \circ }\)
2. Find the value of \(\alpha\)

1. A particle \(P\), of mass 0.5 kg , is moving with velocity ( \(4 \mathbf { i } + 4 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse I of magnitude 2.5 Ns.

As a result of the impulse, the direction of motion of \(P\) is deflected through an angle of \(45 ^ { \circ }\) Given that \(\mathbf { I } = ( \lambda \mathbf { i } + \mu \mathbf { j } )\) Ns, find all the possible pairs of values of \(\lambda\) and \(\mu\).

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]

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 \(m\).

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]

1 Two particles \(P\) and \(Q\) of masses 0.2 kg and 0.3 kg respectively are free to move in a horizontal straight line on a smooth horizontal plane. \(P\) is projected towards \(Q\) with speed \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At the same instant \(Q\) is projected towards \(P\) with speed \(1 \mathrm {~ms} ^ { - 1 } . Q\) comes to rest in the resulting collision. Find the speed of \(P\) after the collision.

1 Two particles \(P\) and \(Q\), of masses 0.2 kg and 0.5 kg respectively, are at rest on a smooth horizontal plane. \(P\) is projected towards \(Q\) with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Write down the momentum of \(P\).
2. After the collision \(P\) continues to move in the same direction with speed \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the speed of \(Q\) after the collision.

7 The diagram shows a cannon fixed to a trolley. The trolley runs on a smooth horizontal track. \includegraphics[max width=\textwidth, alt={}, center]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-8_310_1086_296_520} A driver boards the trolley with two cannon balls. The combined mass of the trolley, driver, cannon and cannon balls is 320 kg . Each cannon ball has a mass of 5 kg . Initially the trolley is at rest. A force of 480 N acts on the trolley in the forward direction for 4 seconds.

1. 1. Calculate the magnitude of the impulse of the force on the trolley.
   2. Calculate the speed of the trolley after the force stops acting. The driver now fires a cannon ball horizontally in the backward direction. The cannon ball and cannon separate at a rate of \(90 \mathrm {~ms} ^ { - 1 }\).
2. Show that, after the firing of the cannon ball, the trolley moves with a speed of \(7.41 \mathrm {~ms} ^ { - 1 }\), correct to \(\mathbf { 3 }\) significant figures. The driver now reverses the direction of the cannon and fires the second cannon ball horizontally in the forward direction. Again, the cannon ball and cannon separate at a rate of \(90 \mathrm {~ms} ^ { - 1 }\).
3. Calculate the overall percentage change in the kinetic energy of the trolley (alone) from before the first cannon ball is fired to after the second is fired, giving your answer correct to \(\mathbf { 2 }\) decimal places. You should make clear whether the change in kinetic energy is a gain or a loss.
4. Give a reason why one of the modelling assumptions that was required in answering parts (a), (b) and (c) may not have been appropriate. \section*{END OF QUESTION PAPER}

A particle of mass 5 kg is moving with velocity \(2\mathbf{i} + 5\mathbf{j}\) m s\(^{-1}\). It receives an impulse of magnitude 15 N s in the direction \(\mathbf{i} + 2\mathbf{j} - 2\mathbf{k}\). Find the velocity of the particle immediately afterwards. [3]

A tennis ball of mass \(0.1\) kg is hit by a racquet. Immediately before being hit, the ball has velocity \(30\mathbf{i}\) m s\(^{-1}\). The racquet exerts an impulse of \((-2\mathbf{i} - 4\mathbf{j})\) N s on the ball. By modelling the ball as a particle, find the velocity of the ball immediately after being hit. [4]

3. \begin{figure}[h] \end{figure} A particle \(P\) of mass 0.5 kg is moving in a straight line with speed \(2.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it receives an impulse of magnitude 3 Ns .
The angle between the direction of motion of \(P\) immediately before receiving the impulse and the line of action of the impulse is \(\alpha\), where \(\tan \alpha = \frac { 4 } { 3 }\), as shown in Figure 2. Find the speed of \(P\) immediately after receiving the impulse.

2. A particle \(P\) of mass \(2 m\) is moving on a rough horizontal plane when it collides directly with a particle \(Q\) of mass \(4 m\) which is at rest on the plane. The speed of \(P\) immediately before the collision is \(3 u\). The speed of \(Q\) immediately after the collision is \(2 u\).

1. Find, in terms of \(u\), the speed of \(P\) immediately after the collision.
2. State clearly the direction of motion of \(P\) immediately after the collision. Following the collision, \(Q\) comes to rest after travelling a distance \(\frac { 6 u ^ { 2 } } { g }\) along the plane. The coefficient of friction between \(Q\) and the plane is \(\mu\).
3. Find the value of \(\mu\).

1 A ball of mass 0.2 kg is travelling horizontally at \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it hits a vertical wall.
It rebounds horizontally at \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Find the magnitude of the impulse exerted on the ball by the wall.
Circle your answer.
[0pt] [1 mark]
0.4 N s
1.4 N s

A block \(A\) of mass 9 kg is released from rest from a point \(P\) which is a height \(h\) metres above horizontal soft ground. The block falls and strikes another block \(B\) of mass 1.5 kg which is on the ground vertically below \(P\). The speed of \(A\) immediately before it strikes \(B\) is 7 m s\(^{-1}\). The blocks are modelled as particles.

1. Find the value of \(h\). [2] Immediately after the impact the blocks move downwards together with the same speed and both come to rest after sinking a vertical distance of 12 cm into the ground. Assuming that the resistance offered by the ground has constant magnitude \(R\) newtons,
2. find the value of \(R\). [8]

Two particles, \(P\) and \(Q\), of mass 2 kg and 1.5 kg respectively are at rest on a smooth, horizontal surface. They are connected by a light, inelastic string which is initially slack. Particle \(P\) is projected away from \(Q\) with a speed of 7 ms\(^{-1}\).

1. Find the common speed of the particles after the string becomes taut. [3 marks]
2. Calculate the impulse in the string when it jerks tight. [2 marks]

1 The graph shows how a force, \(F\) newtons, varies during a 5 second period of time. \includegraphics[max width=\textwidth, alt={}, center]{0afe3ff2-0af5-4aeb-98c5-1346fa803388-02_575_1182_680_429} Calculate the magnitude of the impulse of the force.
Circle your answer.
[0pt] [1 mark]
17.5 N s
25 Ns
35 Ns
70 Ns

1. Two particles \(P\) and \(Q\) have mass 0.4 kg and 0.6 kg respectively. The particles are initially at rest on a smooth horizontal table. Particle \(P\) is given an impulse of magnitude 3 N s in the direction \(P Q\).
   1. Find the speed of \(P\) immediately before it collides with \(Q\).
   Immediately after the collision between \(P\) and \(Q\), the speed of \(Q\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Show that immediately after the collision \(P\) is at rest.

A bullet of mass 0.08 kg is fired horizontally into a fixed vertical barrier. It enters the barrier horizontally with speed 300 m s\(^{-1}\) and emerges horizontally after 0.02 s. There is a constant horizontal resisting force of magnitude 1000 N. Find the speed with which the bullet emerges from the barrier. [3]

1. Determine the impulse of a force of magnitude \(6\) N that acts on a particle of mass \(3\) kg for \(1.5\) seconds. [1]

Particles \(A\) and \(B\), of masses \(0.1\) kg and \(0.2\) kg respectively, can move on a smooth horizontal table. Initially \(A\) is moving with speed \(3\) m s\(^{-1}\) towards \(B\), which is moving with speed \(1\) m s\(^{-1}\) in the same direction as the motion of \(A\). During a collision \(B\) experiences an impulse from \(A\) of magnitude \(0.2\) kg m s\(^{-1}\).

1. Find the speeds of the particles immediately after the collision. [4]
2. Determine the coefficient of restitution between the particles. [2]

\includegraphics{figure_4} Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(2m\) respectively. Sphere \(B\) is at rest on a smooth horizontal surface. Sphere \(A\) is moving on the surface with speed \(u\) at an angle of \(30°\) to the line of centres of \(A\) and \(B\) when it collides with \(B\) (see diagram). The coefficient of restitution between the spheres is \(e\).

1. Show that the speed of \(B\) after the collision is \(\frac{\sqrt{3}}{6}u(1 + e)\) and find the speed of \(A\) after the collision. [6]
2. Given that \(e = \frac{1}{2}\), find the loss of kinetic energy as a result of the collision. [3]

\includegraphics{figure_7} \(PQ\) is a line of greatest slope, of length \(4\) m, on a smooth plane inclined at \(30°\) to the horizontal. Particles \(A\) and \(B\), of masses \(0.15\) kg and \(0.5\) kg respectively, move along \(PQ\) with \(A\) below \(B\). The particles are both moving upwards, \(A\) with speed \(8\) m s\(^{-1}\) and \(B\) with speed \(2\) m s\(^{-1}\), when they collide at the mid-point of \(PQ\) (see diagram). Particle \(A\) is instantaneously at rest immediately after the collision.

1. Show that \(B\) does not reach \(Q\) in the subsequent motion. [8]
2. Find the time interval between the instant of \(A\)'s arrival at \(P\) and the instant of \(B\)'s arrival at \(P\). [6]

6 A particle \(A\) is projected vertically upwards from level ground with an initial speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At the same instant a particle \(B\) is released from rest 15 m vertically above \(A\). The mass of one of the particles is twice the mass of the other particle. During the subsequent motion \(A\) and \(B\) collide and coalesce to form particle \(C\). Find the difference between the two possible times at which \(C\) hits the ground. \(7 \quad\) A particle \(P\) moving in a straight line starts from rest at a point \(O\) and comes to rest 16 s later. At time \(t \mathrm {~s}\) after leaving \(O\), the acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) of \(P\) is given by $$\begin{array} { l l } a = 6 + 4 t & 0 \leqslant t < 2 , \\ a = 14 & 2 \leqslant t < 4 , \\ a = 16 - 2 t & 4 \leqslant t \leqslant 16 . \end{array}$$ There is no sudden change in velocity at any instant.

1. Find the values of \(t\) when the velocity of \(P\) is \(55 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Complete the sketch of the velocity-time diagram. \includegraphics[max width=\textwidth, alt={}, center]{41e63d05-d109-47dc-80a6-927953e3e607-11_511_1054_351_584}
3. Find the distance travelled by \(P\) when it is decelerating.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

Small smooth spheres \(A\) and \(B\), of equal radii and of masses \(5\text{kg}\) and \(3\text{kg}\) respectively, lie on a smooth horizontal plane. Initially \(B\) is at rest and \(A\) is moving towards \(B\) with speed \(8.5\text{ms}^{-1}\). The spheres collide and after the collision \(A\) continues to move in the same direction but with a quarter of the speed of \(B\).

1. Find the speed of \(B\) after the collision. [3]
2. Find the loss of kinetic energy of the system due to the collision. [2]

A light spring \(AB\) has natural length \(a\) and modulus of elasticity \(5mg\). The end \(A\) of the spring is attached to a fixed point on a smooth horizontal surface. A particle \(P\) of mass \(m\) is attached to the end \(B\) of the spring. The spring and particle \(P\) are at rest on the surface. Another particle \(Q\) of mass \(km\) is moving with speed \(\sqrt{4ga}\) along the horizontal surface towards \(P\) in the direction \(BA\). The particles \(P\) and \(Q\) collide directly and coalesce. In the subsequent motion the greatest amount by which the spring is compressed is \(\frac{2}{3}a\). Find the value of \(k\). [6]