Successive collisions, three particles in line

1. Particles \(A , B\) and \(C\) of masses \(4 m , 3 m\) and \(m\) respectively, lie at rest in a straight line on a smooth horizontal plane with \(B\) between \(A\) and \(C\). Particles \(A\) and \(B\) are projected towards each other with speeds \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively, and collide directly.

As a result of the collision, \(A\) is brought to rest and \(B\) rebounds with speed \(k v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 3 } { 4 }\).

1. Show that \(u = 3 v\).
2. Find the value of \(k\). Immediately after the collision between \(A\) and \(B\), particle \(C\) is projected with speed \(2 v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) towards \(B\) so that \(B\) and \(C\) collide directly.
3. Show that there is no further collision between \(A\) and \(B\).

7. Three particles \(P , Q\) and \(R\) lie at rest in a straight line on a smooth horizontal table with \(Q\) between \(P\) and \(R\). The particles \(P , Q\) and \(R\) have masses \(2 m , 3 m\) and \(4 m\) respectively. Particle \(P\) is projected towards \(Q\) with speed \(u\) and collides directly with it. The coefficient of restitution between each pair of particles is \(e\).

1. Show that the speed of \(Q\) immediately after the collision with \(P\) is \(\frac { 2 } { 5 } ( 1 + e ) u\). After the collision between \(P\) and \(Q\) there is a direct collision between \(Q\) and \(R\).
   Given that \(e = \frac { 3 } { 4 }\), find
2. 1. the speed of \(Q\) after this collision,
   2. the speed of \(R\) after this collision. Immediately after the collision between \(Q\) and \(R\), the rate of increase of the distance between \(P\) and \(R\) is \(V\).
3. Find \(V\) in terms of \(u\).

2 Three uniform small smooth spheres, \(A , B\) and \(C\), have equal radii. Their masses are \(4 m , 2 m\) and \(m\) respectively. They lie in a straight line on a smooth horizontal surface with \(B\) between \(A\) and \(C\). Initially \(A\) is moving towards \(B\) with speed \(u , B\) is at rest and \(C\) is moving in the same direction as \(A\) with speed \(\frac { 1 } { 2 } u\). The coefficient of restitution between any two of the spheres is \(e\). The first collision is between \(A\) and \(B\). In this collision sphere \(A\) loses three-quarters of its kinetic energy. Show that \(e = \frac { 1 } { 2 }\). Find the speed of \(B\) after its collision with \(C\) and deduce that there are no further collisions between the spheres.

5 Three uniform small smooth spheres \(A , B\) and \(C\) have equal radii and masses \(3 m , 2 m\) and \(m\) respectively. The spheres are at rest in a straight line on a smooth horizontal surface, with \(B\) between \(A\) and \(C\). The coefficient of restitution between \(A\) and \(B\) is \(e\) and the coefficient of restitution between \(B\) and \(C\) is \(e ^ { \prime }\). Sphere \(A\) is projected directly towards \(B\) with speed \(u\). Show that, after the collision between \(B\) and \(C\), the speed of \(C\) is \(\frac { 2 } { 5 } u ( 1 + e ) \left( 1 + e ^ { \prime } \right)\) and find the corresponding speed of \(B\). After this collision between \(B\) and \(C\) it is found that each of the three spheres has the same momentum. Find the values of \(e\) and \(e ^ { \prime }\).

3 Three uniform small spheres \(A , B\) and \(C\) have equal radii and masses \(3 m , m\) and \(m\) respectively. The spheres are at rest in a straight line on a smooth horizontal surface, with \(B\) between \(A\) and \(C\). The coefficient of restitution between each pair of spheres is \(e\). Sphere \(A\) is projected directly towards \(B\) with speed \(u\).

1. Find, in terms of \(u\) and \(e\), expressions for the speeds of \(A , B\) and \(C\) after the first two collisions.
2. Given that \(A\) and \(C\) are moving with equal speeds after these two collisions, find the value of \(e\). [3] \includegraphics[max width=\textwidth, alt={}, center]{34dd6523-7c0c-4842-bbda-56ad8d3f9766-08_812_520_260_808} An object consists of two hollow spheres which touch each other, together with a thin uniform \(\operatorname { rod } A B\). The rod passes through small holes in the surfaces of the spheres. The rod is fixed to the spheres so that it passes through the centre of the smaller sphere. The end \(B\) of the rod is at the centre of the larger sphere. The larger sphere has radius \(2 a\) and mass \(M\), the smaller sphere has radius \(a\) and mass \(k M\), and the rod has length \(7 a\) and mass \(5 M\). A fixed horizontal axis \(L\) passes through \(A\) and is perpendicular to \(A B\) (see diagram).

4 Three particles \(A , B\) and \(C\) have masses \(m , 2 m\) and \(m\) respectively. The particles are able to move on a smooth horizontal surface in a straight line, and \(B\) is between \(A\) and \(C\). Initially \(A\) is moving towards \(B\) with speed \(2 u\) and \(C\) is moving towards \(B\) with speed \(u\). The particle \(B\) is at rest. The coefficient of restitution between any pair of particles is \(e\). The first collision is between \(A\) and \(B\).

1. Show that the speed of \(B\) immediately before its collision with \(C\) is \(\frac { 2 } { 3 } u ( 1 + e )\).
2. Find the velocity of \(B\) immediately after its collision with \(C\).
3. Given that \(e > \frac { 1 } { 2 }\), show that there are no further collisions between the particles.

2 Three uniform small smooth spheres \(A , B\) and \(C\), of equal radii and of masses \(4 m , \lambda m\) and \(m\) respectively, are at rest in a straight line on a smooth horizontal plane, with \(B\) between \(A\) and \(C\). Sphere \(A\) is projected directly towards \(B\) with speed \(u\). The coefficient of restitution between \(A\) and \(B\), and between \(B\) and \(C\), is \(\frac { 1 } { 2 }\). Show that the speed of \(B\) after it is struck by \(A\) is \(\frac { 6 u } { \lambda + 4 }\). Given that the speed of \(C\) after it is struck by \(B\) is \(u\), find the value of \(\lambda\).

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.

6 Three particles \(A , B\) and \(C\) are free to move in the same straight line on a large smooth horizontal surface. Their masses are \(3.3 \mathrm {~kg} , 2.2 \mathrm {~kg}\) and 1 kg respectively. The coefficient of restitution in collisions between any two of them is \(e\). Initially, \(B\) and \(C\) are at rest and \(A\) is moving towards \(B\) with speed \(u \mathrm {~ms} ^ { - 1 }\) (see diagram). \(A\) collides directly with \(B\) and \(B\) then goes on to collide directly with \(C\). \includegraphics[max width=\textwidth, alt={}, center]{0501e5a4-2137-4e7d-98ff-2ee81941cbf3-4_221_1342_552_246}

1. The velocities of \(A\) and \(B\) immediately after the first collision are denoted by \(\mathrm { v } _ { \mathrm { A } } \mathrm { ms } ^ { - 1 }\) and \(\mathrm { V } _ { \mathrm { B } } \mathrm { ms } ^ { - 1 }\) respectively.
   After the collision between \(B\) and \(C\) there is a further collision between \(A\) and \(B\).
2. Determine the range of possible values of \(e\).

1

1. In this part-question, all the objects move along the same straight line on a smooth horizontal plane. All their collisions are direct. The masses of the objects \(\mathrm { P } , \mathrm { Q }\) and R and the initial velocities of P and Q (but not R ) are shown in Fig. 1.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c8f26b7e-1be1-4abf-8fea-6847185fad81-2_177_1011_488_529} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
   \end{figure} A force of 21 N acts on P for 2 seconds in the direction \(\mathrm { PQ } . \mathrm { P }\) does not reach Q in this time.
   1. Calculate the speed of P after the 2 seconds. The force of 21 N is removed after the 2 seconds. When P collides with Q they stick together (coalesce) to form an object S of mass 6 kg .
   2. Show that immediately after the collision S has a velocity of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) towards R . The collision between S and R is elastic with coefficient of restitution \(\frac { 1 } { 4 }\). After the collision, S has a velocity of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of its motion before the collision.
   3. Find the velocities of R before and after the collision. \section*{(b) In this part-question take \(\boldsymbol { g } = \mathbf { 1 0 }\).} A particle of mass 0.2 kg is projected vertically downwards with initial speed \(5 \mathrm {~ms} ^ { - 1 }\) and it travels 10 m before colliding with a fixed smooth plane. The plane is inclined at \(\alpha\) to the vertical where \(\tan \alpha = \frac { 3 } { 4 }\). Immediately after its collision with the plane, the particle has a speed of \(13 \mathrm {~ms} ^ { - 1 }\). This information is shown in Fig. 1.2. Air resistance is negligible. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{c8f26b7e-1be1-4abf-8fea-6847185fad81-2_383_341_1795_854} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
      \end{figure}
   4. Calculate the angle between the direction of motion of the particle and the plane immediately after the collision. Calculate also the coefficient of restitution in the collision.
   5. Calculate the magnitude of the impulse of the plane on the particle.

2 Three smooth spheres \(A , B\) and \(C\) of equal radii and masses \(m , m\) and \(2 m\) respectively lie at rest on a smooth horizontal table. The centres of the spheres lie in a straight line with \(B\) between \(A\) and \(C\). The coefficient of restitution between any two spheres is \(e\). The sphere \(A\) is projected directly towards \(B\) with speed \(u\) and collides with \(B\).

1. Find, in terms of \(u\) and \(e\), the speed of \(B\) immediately after the impact between \(A\) and \(B\).
2. The sphere \(B\) subsequently collides with \(C\). The speed of \(C\) immediately after this collision is \(\frac { 3 } { 8 } u\). Find the value of \(e\).

3 Three smooth spheres, \(A , B\) and \(C\), of equal radii have masses \(1 \mathrm {~kg} , 3 \mathrm {~kg}\) and \(x \mathrm {~kg}\) respectively. The spheres lie at rest in a straight line on a smooth horizontal surface with \(B\) between \(A\) and \(C\). The sphere \(A\) is projected with speed \(3 u\) directly towards \(B\) and collides with it. \includegraphics[max width=\textwidth, alt={}, center]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-08_250_835_511_605} The coefficient of restitution between each pair of spheres is \(\frac { 1 } { 3 }\).

1. Show that \(A\) is brought to rest by the impact and find the speed of \(B\) immediately after the collision in terms of \(u\).
2. Subsequently, \(B\) collides with \(C\). Show that the speed of \(C\) immediately after the collision is \(\frac { 4 u } { 3 + x }\).
   Find the speed of \(B\) immediately after the collision in terms of \(u\) and \(x\).
3. Show that \(B\) will collide with \(A\) again if \(x > 9\).
4. Given that \(x = 5\), find the magnitude of the impulse exerted on \(C\) by \(B\) in terms of \(u\).
   \includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-09_2484_1709_223_153}
   \includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-10_2484_1712_223_153}
   \includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-11_2484_1709_223_153}

4 Three uniform smooth spheres, \(A , B\) and \(C\), have equal radii and masses \(m , 2 m\) and \(6 m\) respectively. The spheres lie at rest in a straight line on a smooth horizontal surface with \(B\) between \(A\) and \(C\). The sphere \(A\) is projected with speed \(u\) directly towards \(B\) and collides with it. \includegraphics[max width=\textwidth, alt={}, center]{bcd20c69-cace-408c-8961-169c19ff0231-10_218_1164_500_438} The coefficient of restitution between \(A\) and \(B\) is \(\frac { 2 } { 3 }\).

1. 1. Show that the speed of \(B\) immediately after the collision is \(\frac { 5 } { 9 } u\).
   2. Find, in terms of \(u\), the speed of \(A\) immediately after the collision.
2. Subsequently, \(B\) collides with \(C\). The coefficient of restitution between \(B\) and \(C\) is \(e\). Show that \(B\) will collide with \(A\) again if \(e > k\), where \(k\) is a constant to be determined.
3. Explain why it is not necessary to model the spheres as particles in this question.
   [0pt] [2 marks]