6.03k Newton's experimental law: direct impact

2 \includegraphics[max width=\textwidth, alt={}, center]{58728f93-bfdb-4f76-a9b9-3a1d1592bfc9-2_531_760_927_696} Two smooth vertical walls each with their base on a smooth horizontal surface intersect at an angle of \(60 ^ { \circ }\). A small smooth sphere \(P\) is moving on the horizontal surface with speed \(u\) when it collides with the first vertical wall at the point \(D\). The angle between the direction of motion of \(P\) and the wall is \(\alpha ^ { \circ }\) before the collision and \(75 ^ { \circ }\) after the collision. The speed of \(P\) after this collision is \(v\) and the coefficient of restitution between \(P\) and the first wall is \(e\). Sphere \(P\) then collides with the second vertical wall at the point \(E\). The speed of \(P\) after this second collision is \(\frac { 1 } { 4 } u\) (see diagram). The coefficient of restitution between \(P\) and the second wall is \(\frac { 3 } { 4 }\).

1. By considering the collision at \(E\), show that \(v = \frac { \sqrt { } 2 } { 5 } u\).
2. Find the value of \(\alpha\) and the value of \(e\).

4 A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle is held vertically above \(O\) with the string taut and then projected horizontally with speed \(\sqrt { } \left( \frac { 13 } { 3 } a g \right)\). It begins to move in a vertical circle with centre \(O\). When \(P\) is at its lowest point, it collides with a stationary particle of mass \(\lambda m\). The two particles coalesce.

1. Show that the speed of the combined particle immediately after the impact is \(\frac { 5 } { \lambda + 1 } \sqrt { } \left( \frac { 1 } { 3 } a g \right)\). In the subsequent motion, the string becomes slack when the combined particle is at a height of \(\frac { 1 } { 3 } a\) above the level of \(O\).
2. Find the value of \(\lambda\).
3. Find, in terms of \(m\) and \(g\), the instantaneous change in the tension in the string as a result of the collision.

2 \includegraphics[max width=\textwidth, alt={}, center]{62d0d8cb-8f8c-4298-9705-71a735a9a4e7-2_531_760_927_696} Two smooth vertical walls each with their base on a smooth horizontal surface intersect at an angle of \(60 ^ { \circ }\). A small smooth sphere \(P\) is moving on the horizontal surface with speed \(u\) when it collides with the first vertical wall at the point \(D\). The angle between the direction of motion of \(P\) and the wall is \(\alpha ^ { \circ }\) before the collision and \(75 ^ { \circ }\) after the collision. The speed of \(P\) after this collision is \(v\) and the coefficient of restitution between \(P\) and the first wall is \(e\). Sphere \(P\) then collides with the second vertical wall at the point \(E\). The speed of \(P\) after this second collision is \(\frac { 1 } { 4 } u\) (see diagram). The coefficient of restitution between \(P\) and the second wall is \(\frac { 3 } { 4 }\).

1. By considering the collision at \(E\), show that \(v = \frac { \sqrt { } 2 } { 5 } u\).
2. Find the value of \(\alpha\) and the value of \(e\).

2 \includegraphics[max width=\textwidth, alt={}, center]{184020e1-7ff2-4172-8d33-baff963afa76-2_531_760_927_696} Two smooth vertical walls each with their base on a smooth horizontal surface intersect at an angle of \(60 ^ { \circ }\). A small smooth sphere \(P\) is moving on the horizontal surface with speed \(u\) when it collides with the first vertical wall at the point \(D\). The angle between the direction of motion of \(P\) and the wall is \(\alpha ^ { \circ }\) before the collision and \(75 ^ { \circ }\) after the collision. The speed of \(P\) after this collision is \(v\) and the coefficient of restitution between \(P\) and the first wall is \(e\). Sphere \(P\) then collides with the second vertical wall at the point \(E\). The speed of \(P\) after this second collision is \(\frac { 1 } { 4 } u\) (see diagram). The coefficient of restitution between \(P\) and the second wall is \(\frac { 3 } { 4 }\).

1. By considering the collision at \(E\), show that \(v = \frac { \sqrt { } 2 } { 5 } u\).
2. Find the value of \(\alpha\) and the value of \(e\).

3 Three uniform small smooth spheres \(A , B\) and \(C\) have equal radii and masses \(m , k m\) and \(m\) respectively, where \(k\) is a constant. The spheres are moving in the same direction along a straight line on a smooth horizontal surface, with \(B\) between \(A\) and \(C\). The speeds of \(A , B\) and \(C\) are \(2 u , u\) and \(\frac { 4 } { 3 } u\) respectively. The coefficient of restitution between any pair of the spheres is \(\frac { 1 } { 2 }\). After sphere \(A\) has collided with sphere \(B\), sphere \(B\) collides with sphere \(C\).

1. Find an inequality satisfied by \(k\).
2. Given that \(k = 2\), show that after \(B\) has collided with \(C\) there are no further collisions between any of the three spheres.

2 Two uniform small smooth spheres \(A\) and \(B\) have equal radii and masses \(5 m\) and \(2 m\) respectively. Sphere \(A\) is moving with speed \(u\) on a smooth horizontal surface when it collides directly with sphere \(B\) which is moving towards it with speed \(2 u\). The coefficient of restitution between the spheres is \(e\).

1. Show that the speed of \(B\) after the collision is \(\frac { 1 } { 7 } u ( 1 + 15 e )\) and find an expression for the speed of \(A\).
   In the collision, the speed of \(A\) is halved and its direction of motion is reversed.
2. Find the value of \(e\).
3. For this collision, find the ratio of the loss of kinetic energy of \(A\) to the loss of kinetic energy of \(B\). \includegraphics[max width=\textwidth, alt={}, center]{f2073c6e-0f76-4246-89a7-2f3a9f7aaff8-04_630_332_264_900} A uniform disc, of radius \(a\) and mass \(2 M\), is attached to a thin uniform rod \(A B\) of length \(6 a\) and mass \(M\). The rod lies along a diameter of the disc, so that the centre of the disc is a distance \(x\) from \(A\) (see diagram).
4. Find the moment of inertia of the object, consisting of disc and rod, about a fixed horizontal axis \(l\) through \(A\) and perpendicular to the plane of the disc.
   The object is free to rotate about the axis \(l\). The object is held with \(A B\) horizontal and is released from rest. When \(A B\) makes an angle \(\theta\) with the vertical, where \(\cos \theta = \frac { 3 } { 5 }\), the angular speed of the object is \(\sqrt { } \left( \frac { 2 g } { 5 a } \right)\).
5. Find the possible values of \(x\).

2 A small uniform sphere \(A\), of mass \(2 m\), is moving with speed \(u\) on a smooth horizontal surface when it collides directly with a small uniform sphere \(B\), of mass \(m\), which is at rest. The spheres have equal radii and the coefficient of restitution between them is \(e\).

1. Find expressions for the speeds of \(A\) and \(B\) immediately after the collision.
   Subsequently \(B\) collides with a vertical wall which is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the wall is 0.4 . After \(B\) has collided with the wall, the speeds of \(A\) and \(B\) are equal.
2. Find \(e\).
3. Initially \(B\) is at a distance \(d\) from the wall. Find the distance of \(B\) from the wall when it next collides with \(A\). \(3 A\) and \(B\) are two fixed points on a smooth horizontal surface, with \(A B = 3 a \mathrm {~m}\). One end of a light elastic string, of natural length \(a\) m and modulus of elasticity \(m g \mathrm {~N}\), is attached to the point \(A\). The other end of this string is attached to a particle \(P\) of mass \(m \mathrm {~kg}\). One end of a second light elastic string, of natural length \(k a \mathrm {~m}\) and modulus of elasticity \(2 m g \mathrm {~N}\), is attached to \(B\). The other end of this string is attached to \(P\). It is given that the system is in equilibrium when \(P\) is at \(M\), the mid-point of \(A B\).

7. Two particles \(A\) and \(B\), of mass \(2 m\) and \(3 m\) respectively, are initially at rest on a smooth horizontal surface. Particle \(A\) is projected with speed \(3 u\) towards \(B\). Particle \(A\) collides directly with particle \(B\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 3 } { 4 }\)

1. Find
   1. the speed of \(A\) immediately after the collision,
   2. the speed of \(B\) immediately after the collision. After the collision \(B\) hits a fixed smooth vertical wall and rebounds. The wall is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the wall is \(e\). The magnitude of the impulse received by \(B\) when it hits the wall is \(\frac { 27 } { 4 } m u\).
2. Find the value of \(e\).
3. Determine whether there is a further collision between \(A\) and \(B\) after \(B\) rebounds from the wall.
   

4 \includegraphics[max width=\textwidth, alt={}, center]{66eb8290-3a80-40bf-be40-a936ed7d5a1b-3_136_824_260_623} Particles \(P\) and \(Q\) are moving towards each other with constant speeds \(4 \mathrm {~ms} ^ { - 1 }\) and \(2 \mathrm {~ms} ^ { - 1 }\) along the same straight line on a smooth horizontal surface (see diagram). \(P\) has mass 0.2 kg and \(Q\) has mass 0.3 kg . The two particles collide.

1. Show that \(Q\) must change its direction of motion in the collision.
2. Given that \(P\) and \(Q\) move with equal speed after the collision, calculate both possible values for their speed after they collide.

3 A small sphere of mass 0.2 kg is projected vertically downwards with speed \(21 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point at a height of 40 m above horizontal ground. It hits the ground and rebounds vertically upwards, coming to instantaneous rest at its initial point of projection. Ignoring air resistance, calculate

1. the coefficient of restitution between the sphere and the ground,
2. the magnitude of the impulse which the ground exerts on the sphere.

1 A ball is projected with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation of \(55 ^ { \circ }\) above the horizontal. At the instant when the ball reaches its greatest height, it hits a vertical wall, which is perpendicular to the ball's path. The coefficient of restitution between the ball and the wall is 0.65 . Calculate the speed of the ball

1. immediately before its impact with the wall,
2. immediately after its impact with the wall.

5 A particle \(P\) of mass \(2 m\) is moving on a smooth horizontal surface with speed \(u\) when it collides directly with a particle \(Q\) of mass \(k m\) whose speed is \(3 u\) in the opposite direction. As a result of the collision, the directions of motion of both particles are reversed and the speed of \(P\) is halved.

1. Find, in terms of \(u\) and \(k\), the speed of \(Q\) after the collision. Hence write down the range of possible values of \(k\).
2. Calculate the magnitude of the impulse which \(Q\) exerts on \(P\).
3. Given that \(k = \frac { 1 } { 2 }\), calculate the coefficient of restitution between \(P\) and \(Q\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{982647bd-8514-40cf-b4ee-674f51df32c5-3_472_1143_221_242} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} One end of a light inextensible string is attached to a point \(P\). The other end is attached to a point \(Q , 1.96 \mathrm {~m}\) vertically below \(P\). A small smooth bead \(B\), of mass 0.3 kg , is threaded on the string and moves in a horizontal circle with centre \(Q\) and radius \(1.96 \mathrm {~m} . B\) rotates about \(Q\) with constant angular speed \(\omega\) rad s \(^ { - 1 }\) (see Fig. 1).

6 A particle is projected from a point \(O\) with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation \(\theta\) above the horizontal and it moves freely under gravity. The horizontal and upward vertical displacements of the particle from \(O\) at any subsequent time, \(t\) seconds, are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.

1. Express \(x\) and \(y\) in terms of \(\theta\) and \(t\), and hence show that $$y = x \tan \theta - \frac { 4.9 x ^ { 2 } } { v ^ { 2 } \cos ^ { 2 } \theta } .$$
   \includegraphics[max width=\textwidth, alt={}]{dd23f4a8-f7e7-4f80-bad6-7e9ec21565fc-4_551_575_667_826}
   The particle subsequently passes through the point \(A\) with coordinates \(( h , - h )\) as shown in the diagram. It is given that \(v = 14\) and \(\theta = 30 ^ { \circ }\).
2. Calculate \(h\).
3. Calculate the direction of motion of the particle at \(A\).
4. Calculate the speed of the particle at \(A\). \includegraphics[max width=\textwidth, alt={}, center]{dd23f4a8-f7e7-4f80-bad6-7e9ec21565fc-4_278_1061_1749_543} Two small spheres, \(P\) and \(Q\), are free to move on the inside of a smooth hollow cylinder, in such a way that they remain in contact with both the curved surface and the base of the cylinder. The mass of \(P\) is 0.2 kg , the mass of \(Q\) is 0.3 kg and the radius of the cylinder is \(0.4 \mathrm {~m} . P\) and \(Q\) are stationary at opposite ends of a diameter of the base of the cylinder (see diagram). The coefficient of restitution between \(P\) and \(Q\) is \(0.5 . P\) is given an impulse of magnitude 0.8 Ns in a tangential direction.
5. Calculate the speeds of the particles after \(P\) 's first impact with \(Q\). \(Q\) subsequently catches up with \(P\) and there is a second impact.
6. Calculate the speeds of the particles after this second impact.
7. Calculate the magnitude of the force exerted on \(Q\) by the curved surface of the cylinder after the second impact.

2 A small sphere of mass 0.2 kg is dropped from rest at a height of 3 m above horizontal ground. It falls vertically, hits the ground and rebounds vertically upwards, coming to instantaneous rest at a height of 1.8 m above the ground.

1. Calculate the magnitude of the impulse which the ground exerts on the sphere.
2. Calculate the coefficient of restitution between the sphere and the ground.

6 A small ball of mass 0.5 kg is held at a height of 3.136 m above a horizontal floor. The ball is released from rest and rebounds from the floor. The coefficient of restitution between the ball and floor is \(e\).

1. Find in terms of \(e\) the speed of the ball immediately after the impact with the floor and the impulse that the floor exerts on the ball. The ball continues to bounce until it eventually comes to rest.
2. Show that the time between the first bounce and the second bounce is \(1.6 e\).
3. Write down, in terms of \(e\), the time between
   1. the second bounce and the third bounce,
   2. the third bounce and the fourth bounce.
   3. Given that the time from the ball being released until it comes to rest is 5 s , find the value of \(e\).

4 \includegraphics[max width=\textwidth, alt={}, center]{35477eb8-59e0-4de6-889c-1f5841f65eec-3_168_956_246_593} Three smooth spheres \(A , B\) and \(C\), of equal radius and of masses \(m \mathrm {~kg} , 2 m \mathrm {~kg}\) and \(3 m \mathrm {~kg}\) respectively, lie in a straight line and are free to move on a smooth horizontal table. Sphere \(A\) is moving with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it collides directly with sphere \(B\) which is stationary. As a result of the collision \(B\) starts to move with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the coefficient of restitution between \(A\) and \(B\).
2. Find, in terms of \(m\), the magnitude of the impulse that \(A\) exerts on \(B\), and state the direction of this impulse. Sphere \(B\) subsequently collides with sphere \(C\) which is stationary. As a result of this impact \(B\) and \(C\) coalesce.
3. Show that there will be another collision.

8 Two uniform smooth spheres, \(A\) and \(B\), have the same radius. The mass of \(A\) is 2 kg and the mass of \(B\) is \(m \mathrm {~kg}\). Sphere \(A\) is travelling in a straight line on a smooth horizontal surface, with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when it collides directly with sphere \(B\), which is at rest. As a result of the collision, sphere \(A\) continues in the same direction with a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the greatest possible value of \(m\). It is given that \(m = 1\).
2. Find the coefficient of restitution between \(A\) and \(B\). On another occasion \(A\) and \(B\) are travelling towards each other, each with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when they collide directly.
3. Find the kinetic energy lost due to the collision.

5 Two small smooth discs, \(C\) and \(D\), have equal radii and masses of 2 kg and 3 kg respectively. The discs are sliding on a smooth horizontal surface towards each other and collide directly. Disc \(C\) has speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and disc \(D\) has speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) as they collide. The coefficient of restitution between \(C\) and \(D\) is 0.6 The diagram shows the discs, viewed from above, before the collision. \includegraphics[max width=\textwidth, alt={}, center]{18522f4c-4aa2-4ef5-898f-5ad2b06e287c-06_343_712_868_753} 5

1. Show that the speed of \(D\) immediately after the collision is \(1.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 2 significant figures.
   5
2. Find the speed of \(C\) immediately after the collision.
   [0pt] [2 marks]
   5
3. In fact the horizontal surface on which the discs are sliding is not smooth.
   Explain how the introduction of friction will affect your answer to part (b).
   [0pt] [2 marks]
   Turn over for the next question

4 \includegraphics[max width=\textwidth, alt={}, center]{5960a9cf-2c51-4c07-9973-c29604762df7-3_218_1335_251_367} Three particles \(A\), \(B\) and \(C\) are free to move in the same straight line on a large smooth horizontal surface. Their masses are \(1.2 \mathrm {~kg} , 1.8 \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively (see diagram). The coefficient of restitution in collisions between any two of them is \(\frac { 3 } { 4 }\). Initially, \(B\) and \(C\) are at rest and \(A\) is moving with a velocity of \(4.0 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) towards \(B\).

1. Show that immediately after the collision between \(A\) and \(B\) the speed of \(B\) is \(2.8 \mathrm {~ms} ^ { - 1 }\).
2. Find the velocity of \(A\) immediately after this collision. \(B\) subsequently collides with \(C\).
3. Find, in terms of \(m\), the velocity of \(B\) after its collision with \(C\).
4. Given that the direction of motion of \(B\) is reversed by the collision with \(C\), find the range of possible values of \(m\).

6 Particles \(A\) of mass \(2 m\) and \(B\) of mass \(m\) are on a smooth horizontal floor. \(A\) is moving with speed \(u\) directly towards a vertical wall, and \(B\) is at rest between \(A\) and the wall (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{74bada9e-60cf-4ed4-abd0-4be155b7cf81-5_224_828_354_244} A collides directly with \(B\). The coefficient of restitution in this collision is \(\frac { 1 } { 2 }\). \(B\) then collides with the wall, rebounds, and collides with \(A\) for a second time.

1. Show that the speed of \(B\) after its second collision with \(A\) is \(\frac { 1 } { 2 } u\). The first collision between \(A\) and \(B\) occurs at a distance \(d\) from the wall. The second collision between \(A\) and \(B\) occurs at a distance \(\frac { 1 } { 5 } d\) from the wall.
2. Find the coefficient of restitution for the collision between \(B\) and the wall.

1 Two stones, A and B , are sliding along the same straight line on a horizontal sheet of ice. Stone A, of mass 50 kg , is moving with a constant velocity of \(2.1 \mathrm {~ms} ^ { - 1 }\) towards stone B. Stone B, of mass 70 kg , is moving with a constant velocity of \(0.8 \mathrm {~ms} ^ { - 1 }\) towards stone A. A and B collide directly. Immediately after their collision stone A's velocity is \(0.35 \mathrm {~ms} ^ { - 1 }\) in the same direction as its velocity before the collision.

1. Find the speed of stone B immediately after the collision.
2. Find the coefficient of restitution for the collision.
3. Find the total loss of kinetic energy caused by the collision.
4. Explain whether the collision was perfectly elastic.

7 Two particles, \(P\) and \(Q\), are on a smooth horizontal floor. \(P\), of mass 1 kg , is moving with speed \(1.79 \mathrm {~ms} ^ { - 1 }\) directly towards a vertical wall. \(Q\), of mass 2.74 kg , is between \(P\) and the wall, moving directly towards \(P\) with speed \(0.08 \mathrm {~ms} ^ { - 1 }\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{98053e88-1aec-4b0d-ae5f-ece4ad340266-4_232_830_1370_246} \(P\) and \(Q\) collide directly and the coefficient of restitution for this collision is denoted by \(e\).

1. Show that after this collision the speed of \(Q\) is given by \(0.42 + 0.5 e \mathrm {~ms} ^ { - 1 }\). After this collision, \(Q\) then goes on to collide directly with the wall. The coefficient of restitution for the collision between \(Q\) and the wall is also \(e\). There is then no subsequent collision between \(P\) and \(Q\).
2. Determine the range of possible values of \(e\).

5 Two identical spheres, \(A\) and \(B\), each of mass 4 kg , are moving directly towards each other along the same straight line on a smooth horizontal surface until they collide. Before they collide, the speeds of \(A\) and \(B\) are \(5 \mathrm {~ms} ^ { - 1 }\) and \(3 \mathrm {~ms} ^ { - 1 }\) respectively. Immediately after they collide, the speed of \(A\) is \(2 \mathrm {~ms} ^ { - 1 }\) and its direction of motion has been reversed.

1. 1. Determine the velocity of \(B\) immediately after \(A\) and \(B\) collide.
   2. Show that the coefficient of restitution between \(A\) and \(B\) is \(\frac { 3 } { 4 }\).
   3. Calculate the total loss of kinetic energy due to this collision. Sphere \(B\) goes on to strike a fixed wall directly. As a result of this collision \(B\) moves along the same straight line with a speed of \(4 \mathrm {~ms} ^ { - 1 }\).
2. Find the coefficient of restitution between \(B\) and the wall, stating whether the collision between \(B\) and the wall is perfectly elastic.
3. Determine the magnitude of the impulse that \(B\) exerts on \(A\) the next time that they collide.

5 Two particles, \(A\) of mass \(m _ { A } \mathrm {~kg}\) and \(B\) of mass 5 kg , are moving directly towards each other on a smooth horizontal floor. Before they collide they have speeds \(\mathrm { u } _ { \mathrm { A } } \mathrm { m } \mathrm { s } ^ { - 1 }\) and \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. Immediately after they collide the direction of motion of each particle has been reversed and \(A\) and \(B\) have speeds \(3.25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively (see diagram). The coefficient of restitution between \(A\) and \(B\) is 0.75 . Before: \includegraphics[max width=\textwidth, alt={}, center]{d2156252-71f2-4084-89a2-4d246583eb65-4_218_711_552_283} After: \includegraphics[max width=\textwidth, alt={}, center]{d2156252-71f2-4084-89a2-4d246583eb65-4_218_707_552_1078}

1. Determine the value of \(m _ { A }\) and the value of \(u _ { A }\).
   [0pt] [5]
2. Show that approximately \(41 \%\) of the kinetic energy of the system is lost in this collision. After the collision between \(A\) and \(B\), \(B\) goes on to collide directly with a third particle \(C\) of mass 3 kg which is travelling towards \(B\) with a speed of \(5.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of restitution between \(B\) and \(C\) is denoted by \(e\).
3. Given that, after \(B\) and \(C\) collide, there are no further collisions between \(A , B\) and \(C\) determine the range of possible values of \(e\).

4 A particle \(P\) of mass 2.4 kg is moving in a straight line \(O A\) on a horizontal plane. \(P\) is acted on by a force of magnitude 30 N in the direction of motion. The distance \(O A\) is 10 m .

1. Find the work done by this force as \(P\) moves from \(O\) to \(A\). The motion of \(P\) is resisted by a constant force of magnitude \(R \mathrm {~N}\). The velocity of \(P\) increases from \(12 \mathrm {~ms} ^ { - 1 }\) at \(O\) to \(18 \mathrm {~ms} ^ { - 1 }\) at \(A\).
2. Find the value of \(R\).
3. Find the average power used in overcoming the resistance force on \(P\) as it moves from \(O\) to \(A\). When \(P\) reaches \(A\) it collides directly with a particle \(Q\) of mass 1.6 kg which was at rest at \(A\) before the collision. The impulse exerted on \(Q\) by \(P\) as a result of the collision is 17.28 Ns .
4. 1. Find the speed of \(Q\) after the collision.
   2. Hence show that the collision is inelastic.