6.03b Conservation of momentum: 1D two particles

2 Three particles, \(A , B\) and \(C\), of masses \(2 \mathrm {~kg} , 3 \mathrm {~kg}\) and 5 kg respectively, are at rest in a straight line on a smooth horizontal plane with \(B\) between \(A\) and \(C\). Collisions between \(A\) and \(B\) are perfectly elastic. The coefficient of restitution for collisions between \(B\) and \(C\) is \(e\). \(A\) is projected towards \(B\) with a speed of \(5 u \mathrm {~ms} ^ { - 1 }\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{709f3a7a-d857-4813-98ab-de6b41a3a8dc-02_190_885_1151_260} Show that only two collisions occur.

1. Two smooth spheres \(A\) and \(B\) are moving on a smooth horizontal plane when they collide obliquely. When the spheres collide, the line joining their centres is parallel to the vector \(\mathbf { j }\), as shown in the diagram below.

Immediately before the collision, sphere \(A\) has velocity ( \(6 \mathbf { i } - 3 \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\) and sphere \(B\) has velocity \(( - 4 \mathbf { i } + 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). Sphere \(A\) has mass 6 kg and sphere \(B\) has mass 2 kg . \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-02_595_972_753_534} Immediately after the collision, sphere \(B\) has velocity \(( - 4 \mathbf { i } - 5 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).

1. Find the velocity of \(A\) immediately after the collision.
   
2. Calculate the coefficient of restitution between \(A\) and \(B\).
   
3. Find the angle through which the direction of motion of \(B\) is deflected as a result of the collision. Give your answer correct to the nearest degree.
   
4. After the collision, sphere \(B\) continues to move with velocity \(( - 4 \mathbf { i } - 5 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) until it collides with another sphere \(C\), which exerts an impulse of \(( - 20 \mathbf { i } + 18 \mathbf { j } )\) Ns on \(B\). Find the velocity of \(B\) after the collision with \(C\).
   \section*{PLEASE DO NOT WRITE ON THIS PAGE}

1. Particle \(A\) has mass \(4 m\) and particle \(B\) has mass \(3 m\).

The particles are moving in opposite directions along the same straight line on a smooth horizontal surface when they collide directly. Immediately before the collision, the speed of \(A\) is \(2 x\) and the speed of \(B\) is \(x\).
Immediately after the collision, the speed of \(A\) is \(y\) and the speed of \(B\) is \(5 y\).
The direction of motion of each particle is reversed as a result of the collision.

1. Show that \(y = \frac { 5 } { 11 } x\).
2. Find, in terms of \(m\) and \(x\), the magnitude of the impulse received by \(A\) in the collision.

1. A particle \(P\) has mass \(5 m\) and a particle \(Q\) has mass \(2 m\).

The particles are moving in opposite directions along the same straight line on a smooth horizontal surface.
Particle \(P\) collides directly with particle \(Q\).
Immediately before the collision, the speed of \(P\) is \(2 u\) and the speed of \(Q\) is \(3 u\). Immediately after the collision, the speed of \(P\) is \(x\) and the speed of \(Q\) is \(y\).
The direction of motion of \(Q\) is reversed as a result of the collision.
The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Find the set of values of \(e\) for which the direction of motion of \(P\) is unchanged as a result of the collision. In the collision, \(Q\) receives an impulse of magnitude \(\frac { 60 } { 7 } m u\)
2. Show that \(e = \frac { 1 } { 5 }\) After the collision, \(Q\) hits a smooth fixed vertical wall that is perpendicular to the direction of motion of \(Q\). Particle \(Q\) rebounds and there is a second collision between \(P\) and \(Q\).
   The coefficient of restitution between \(Q\) and the wall is \(\frac { 1 } { 3 }\)
3. Find, in terms of \(m\) and \(u\), the magnitude of the impulse received by \(Q\) in the second collision between \(P\) and \(Q\).

6 Two smooth spheres, \(A\) and \(B\), have masses \(m\) and \(2 m\) respectively and equal radii. Sphere \(B\) is at rest on a smooth horizontal floor. Sphere \(A\) is projected with speed \(u\) along the floor in a direction parallel to a smooth vertical wall and strikes \(B\) obliquely. Subsequently \(B\) strikes the wall at an angle \(\alpha\) with the wall. The coefficient of restitution between \(A\) and \(B\) and between \(B\) and the wall is 0.5. After \(B\) has struck the wall, \(A\) and \(B\) are moving parallel to each other.

1. Write down a momentum equation and a restitution equation along the line of centres for the impact between \(A\) and \(B\). Hence find the components of velocity of \(A\) and \(B\) in this direction after this first impact.
2. Find the value of \(\alpha\), giving your answer in degrees.

10 \includegraphics[max width=\textwidth, alt={}, center]{d8ca5464-435f-45e0-8e19-1830415a7c60-5_432_949_258_598} A smooth sphere \(P\) of mass \(3 m\) is at rest on a smooth horizontal table. A second smooth sphere \(Q\) of mass \(m\) and the same radius as \(P\) is moving along the table towards \(P\) and strikes it obliquely (see diagram). After the collision, the directions of motion of the two spheres are perpendicular.

1. Find the coefficient of restitution.
2. Given that one sixth of the original kinetic energy is lost as a result of the collision, find the angle between the initial direction of motion of \(Q\) and the line of centres.

10 \includegraphics[max width=\textwidth, alt={}, center]{f0c32e07-f3a0-4d58-bd00-c266177ceaac-4_81_949_1283_598} Three particles \(A , B\) and \(C\), having masses \(1 \mathrm {~kg} , 2 \mathrm {~kg}\) and 5 kg , respectively, are placed 1 metre apart in a straight line on a smooth horizontal plane (see diagram). The particles \(B\) and \(C\) are initially at rest and \(A\) is moving towards \(B\) with speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of restitution between each pair of particles is 0.5 .

1. Find the velocity of \(B\) immediately after the first impact and show that \(A\) comes to rest.
2. Show that \(B\) reverses direction after an impact with \(C\).
3. Find the distance between \(B\) and \(C\) at the instant that \(B\) collides with \(A\) for the second time.

7 A particle \(A\) of mass \(4 m\), on a smooth horizontal plane, is moving with speed \(u\) directly towards another particle \(B\), of mass \(2 m\), which is at rest. The coefficient of restitution between the two particles is \(e\).

1. Show that, after the collision, the velocity of \(A\) is \(\frac { 1 } { 3 } ( 2 - e ) u\) and find the velocity of \(B\).
2. Hence write down their velocities in the case when \(e = \frac { 1 } { 2 }\). Particle \(B\) now collides directly with a third particle \(C\), of mass \(m\), which is at rest. The coefficient of restitution in both collisions is \(\frac { 1 } { 2 }\).
3. Use your answers to part (ii) to find the velocities of \(A , B\) and \(C\) after the second collision has taken place.
4. Explain briefly whether any further collisions take place.

9 A particle of mass 0.5 kg moving on a smooth horizontal plane with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) collides directly with another particle of mass \(k \mathrm {~kg}\) (where \(k\) is a constant) which is at rest. After the collision the first particle comes to rest but the second particle moves off with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find \(v\) in terms of \(k\) and \(u\).
2. The coefficient of restitution between the two particles is \(e\). Find \(e\) in terms of \(k\) only.
3. Show that \(k \geqslant \frac { 1 } { 2 }\).

10 \includegraphics[max width=\textwidth, alt={}, center]{a19fab61-da1c-4803-9dbc-38d618a0c58e-5_435_951_1528_557} A smooth sphere \(P\) of mass \(3 m\) is at rest on a smooth horizontal table. A second smooth sphere \(Q\) of mass \(m\) and the same radius as \(P\) is moving along the table towards \(P\) and strikes it obliquely (see diagram). After the collision, the directions of motion of the two spheres are perpendicular.

1. Find the coefficient of restitution.
2. Given that one-sixth of the original kinetic energy is lost as a result of the collision, find the angle between the initial direction of motion of \(Q\) and the line of centres.

9 \includegraphics[max width=\textwidth, alt={}, center]{01bd6354-3514-4dad-901b-7ecbe155b2c7-5_118_851_1265_607} Three particles \(A , B\) and \(C\), having masses of \(1 \mathrm {~kg} , 2 \mathrm {~kg}\) and 5 kg respectively, are placed 1 metre apart in a straight line on a smooth horizontal plane (see diagram). The particles \(B\) and \(C\) are initially at rest and \(A\) is moving towards \(B\) with speed \(14 \mathrm {~ms} ^ { - 1 }\). The coefficient of restitution between each pair of particles is 0.5 .

1. Find the velocity of \(B\) immediately after the first impact and show that \(A\) comes to rest.
2. Show that \(B\) reversed direction after the impact with \(C\).
3. Find the distances between \(B\) and \(C\) at the instant that \(B\) collides with \(A\) for the second time.

9 \includegraphics[max width=\textwidth, alt={}, center]{b18b1bc5-bf26-4161-b5a5-764b00e97bea-5_118_851_1265_607} Three particles \(A , B\) and \(C\), having masses of \(1 \mathrm {~kg} , 2 \mathrm {~kg}\) and 5 kg respectively, are placed 1 metre apart in a straight line on a smooth horizontal plane (see diagram). The particles \(B\) and \(C\) are initially at rest and \(A\) is moving towards \(B\) with speed \(14 \mathrm {~ms} ^ { - 1 }\). The coefficient of restitution between each pair of particles is 0.5 .

1. Find the velocity of \(B\) immediately after the first impact and show that \(A\) comes to rest.
2. Show that \(B\) reversed direction after the impact with \(C\).
3. Find the distances between \(B\) and \(C\) at the instant that \(B\) collides with \(A\) for the second time.

11 \includegraphics[max width=\textwidth, alt={}, center]{c4bbba86-2968-4247-b300-357217cf213b-5_432_949_909_557} A smooth sphere \(P\) of mass \(3 m\) is at rest on a smooth horizontal table. A second smooth sphere \(Q\) of mass \(m\) and the same radius as \(P\) is moving along the table towards \(P\) and strikes it obliquely (see diagram). After the collision, the directions of motion of the two spheres are perpendicular.

1. Find the coefficient of restitution.
2. Given that one-sixth of the original kinetic energy is lost as a result of the collision, find the angle between the initial direction of motion of \(Q\) and the line of centres.

9 A particle \(P\) of mass \(m \mathrm {~kg}\) is moving with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line on a smooth horizontal table. \(P\) collides directly with a stationary particle \(Q\) of mass 0.5 kg . This collision reverses the direction of motion of \(P\). Immediately after the collision the speed of \(P\) is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(Q\) is \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the value of \(m\),
2. the coefficient of restitution between the two particles.

11 \includegraphics[max width=\textwidth, alt={}, center]{adf5bd3c-5408-421d-b7d5-dea2d0f0185b-6_438_951_255_559} A smooth sphere \(P\) of mass \(3 m\) is at rest on a smooth horizontal table. A second smooth sphere \(Q\) of mass \(m\) and the same radius as \(P\) is moving along the table towards \(P\) and strikes it obliquely (see diagram). After the collision, the directions of motion of the two spheres are perpendicular.

1. Find the coefficient of restitution.
2. Given that one-sixth of the original kinetic energy is lost as a result of the collision, find the angle between the initial direction of motion of \(Q\) and the line of centres.

11 \includegraphics[max width=\textwidth, alt={}, center]{f4acd242-eb78-4124-bfa2-fdecaa188690-6_438_953_264_557} A smooth sphere \(P\) of mass \(3 m\) is at rest on a smooth horizontal table. A second smooth sphere \(Q\) of mass \(m\) and the same radius as \(P\) is moving along the table towards \(P\) and strikes it obliquely (see diagram). After the collision, the directions of motion of the two spheres are perpendicular.

1. Find the coefficient of restitution.
2. Given that one-sixth of the original kinetic energy is lost as a result of the collision, find the angle between the initial direction of motion of \(Q\) and the line of centres.

\includegraphics{figure_3} Two perfectly elastic small smooth spheres \(A\) and \(B\) have masses \(3m\) and \(m\) respectively. They lie at rest on a smooth horizontal plane with \(B\) at a distance \(a\) from a smooth vertical barrier. The line of centres of the spheres is perpendicular to the barrier, and \(B\) is between \(A\) and the barrier (see diagram). Sphere \(A\) is projected towards sphere \(B\) with speed \(u\) and, after the collision between the spheres, \(B\) hits the barrier. The coefficient of restitution between \(B\) and the barrier is \(\frac{1}{4}\). Find the speeds of \(A\) and \(B\) immediately after they first collide, and the distance from the barrier of the point where they collide for the second time. [9]

Two uniform small smooth spheres \(A\) and \(B\) have equal radii and masses \(3m\) and \(m\) respectively. Sphere \(A\) is moving with speed \(u\) on a smooth horizontal surface when it collides directly with sphere \(B\) which is at rest. The coefficient of restitution between the spheres is \(e\).

1. Find, in terms of \(u\) and \(e\), expressions for the velocities of \(A\) and \(B\) after the collision. [3]

Sphere \(B\) continues to move until it strikes a fixed smooth vertical barrier which is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the barrier is \(\frac{3}{4}\). When the spheres subsequently collide, \(A\) is brought to rest.

1. Find the value of \(e\). [7]

Two uniform small smooth spheres \(A\) and \(B\) have equal radii and masses \(3m\) and \(m\) respectively. Sphere \(A\) is moving with speed \(u\) on a smooth horizontal surface when it collides directly with sphere \(B\) which is at rest. The coefficient of restitution between the spheres is \(e\).

1. Find, in terms of \(u\) and \(e\), expressions for the velocities of \(A\) and \(B\) after the collision. [3]
2. Sphere \(B\) continues to move until it strikes a fixed smooth vertical barrier which is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the barrier is \(\frac{3}{4}\). When the spheres subsequently collide, \(A\) is brought to rest. Find the value of \(e\). [7]

Two uniform small smooth spheres \(A\) and \(B\) have equal radii and each has mass \(m\). Sphere \(A\) is moving with speed \(u\) on a smooth horizontal surface when it collides directly with sphere \(B\) which is at rest. The coefficient of restitution between the spheres is \(\frac{2}{3}\). Sphere \(B\) is initially at a distance \(d\) from a fixed smooth vertical wall which is perpendicular to the direction of motion of \(A\). The coefficient of restitution between \(B\) and the wall is \(\frac{1}{3}\).

1. Show that the speed of \(B\) after its collision with the wall is \(\frac{5}{18}u\). [4]
2. Find the distance of \(B\) from the wall when it collides with \(A\) for the second time. [6]

Three uniform small spheres \(A\), \(B\) and \(C\) have equal radii and masses \(2m\), \(4m\) and \(m\) respectively. The spheres are moving 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 \(e\). Spheres \(A\) and \(B\) are moving towards each other with speeds \(2u\) and \(u\) respectively. The first collision is between \(A\) and \(B\).

1. Find the velocities of \(A\) and \(B\) after this collision. [3] Sphere \(C\) is moving towards \(B\) with speed \(\frac{1}{2}u\) and now collides with it. As a result of this collision, \(B\) is brought to rest.
2. Find the value of \(e\). [4]
3. Find the total kinetic energy lost by the three spheres as a result of the two collisions. [3]

Answer only one of the following two alternatives. **EITHER** A particle \(P\), of mass \(m\), is able to move in a vertical circle on the smooth inner surface of a sphere with centre \(O\) and radius \(a\). Points \(A\) and \(B\) are on the inner surface of the sphere and \(AOB\) is a horizontal diameter. Initially, \(P\) is projected vertically downwards with speed \(\sqrt{\left(\frac{21}{2}ag\right)}\) from \(A\) and begins to move in a vertical circle. At the lowest point of its path, vertically below \(O\), the particle \(P\) collides with a stationary particle \(Q\), of mass \(4m\), and rebounds. The speed acquired by \(Q\), as a result of the collision, is just sufficient for it to reach the point \(B\).

1. Find the speed of \(P\) and the speed of \(Q\) immediately after their collision. [7] In its subsequent motion, \(P\) loses contact with the inner surface of the sphere at the point \(D\), where the angle between \(OD\) and the upward vertical through \(O\) is \(\theta\).
2. Find \(\cos \theta\). [5] **OR** A farmer grows two different types of cherries, Type A and Type B. He assumes that the masses of each type are normally distributed. He chooses a random sample of 8 cherries of Type A. He finds that the sample mean mass is 15.1 g and that a 95% confidence interval for the population mean mass, \(\mu\) g, is \(13.5 \leqslant \mu \leqslant 16.7\).

1. Find an unbiased estimate for the population variance of the masses of cherries of Type A. [3] The farmer now chooses a random sample of 6 cherries of Type B and records their masses as follows. $$12.2 \quad 13.3 \quad 16.4 \quad 14.0 \quad 13.9 \quad 15.4$$
2. Test at the 5% significance level whether the mean mass of cherries of Type B is less than the mean mass of cherries of Type A. You should assume that the population variances for the two types of cherry are equal. [9]

Two small smooth spheres \(A\) and \(B\) of equal radius have masses \(m\) and \(3m\) respectively. They lie at rest on a smooth horizontal plane with their line of centres perpendicular to a smooth fixed vertical barrier wall \(9\) feet away from the barrier. The coefficient of restitution between \(A\) and \(B\), and between \(B\) and the barrier, is \(e\), where \(e > \frac{1}{4}\). Sphere \(A\) is projected directly towards \(B\) with speed \(u\). Show that after colliding with \(B\) the direction of motion of \(A\) is reversed. [5] After the impact, \(B\) hits the barrier and rebounds. Show that \(B\) will subsequently collide with \(A\) again unless \(e = 1\). [3]

A small uniform sphere \(A\), of mass \(2m\), 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 \(e\). Find expressions for the speeds of \(A\) and \(B\) immediately after the collision. [4] 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. Find \(e\). [2] Initially \(B\) is at a distance \(d\) from the wall. Find the distance of \(B\) from the wall when it next collides with \(A\). [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 \(O\). When \(P\) is hanging at rest vertically below \(O\), it is projected horizontally. In the subsequent motion \(P\) completes a vertical circle. The speed of \(P\) when it is at its highest point is \(u\). Show that the least possible value of \(u\) is \(\sqrt{(ag)}\). [2] It is now given that \(u = \sqrt{(ag)}\). When \(P\) passes through the lowest point of its path, it collides with, and coalesces with, a stationary particle of mass \(\frac{1}{4}m\). Find the speed of the combined particle immediately after the collision. [4] In the subsequent motion, when \(OP\) makes an angle \(\theta\) with the upward vertical the tension in the string is \(T\). Find an expression for \(T\) in terms of \(m\), \(g\) and \(\theta\). [5] Find the value of \(\cos \theta\) when the string becomes slack. [2]