6.03k Newton's experimental law: direct impact

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.

12 \includegraphics[max width=\textwidth, alt={}, center]{742ef62b-bd72-45b4-88e3-70399632e9d6-4_247_801_1535_671} A small smooth sphere is projected from a point \(A\) across a smooth horizontal surface. The sphere strikes a smooth vertical wall at the point \(P\). The acute angle between the direction of motion of the sphere and the wall is \(\theta\). After the impact, the sphere passes through the point \(B\), where angle \(A P B = \phi\) (see diagram). The coefficient of restitution between the sphere and the wall is \(e\).

1. Given that \(\theta = \tan ^ { - 1 } 3\) and \(\phi = 90 ^ { \circ }\), find the exact value of \(e\).
2. Given instead that \(e = \frac { 2 } { 3 }\) and \(\phi = 45 ^ { \circ }\), show that \(\theta = \tan ^ { - 1 } 3\).

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.

11 A smooth sphere of mass 2 kg has velocity \(( 24 \mathbf { i } - 7 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and is travelling on a horizontal plane, where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in the horizontal plane. The sphere strikes a vertical wall. The line of intersection of the wall and the plane is in the direction \(( 4 \mathbf { i } + 3 \mathbf { j } )\).

1. Show that the acute angle between the path of the sphere before the impact and the direction of the wall is \(\tan ^ { - 1 } \left( \frac { 4 } { 3 } \right)\).
2. After the impact, the velocity of the sphere is \(( 7.2 \mathbf { i } + 15.4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find
   1. the coefficient of restitution between the sphere and the wall,
   2. the magnitude of the impulse exerted by the sphere on the wall.

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 \(k \geqslant \frac { 1 } { 2 }\).

12 \includegraphics[max width=\textwidth, alt={}, center]{1a89caec-6da8-4b83-9ffa-efc209ecbc8d-5_205_200_264_497} \includegraphics[max width=\textwidth, alt={}, center]{1a89caec-6da8-4b83-9ffa-efc209ecbc8d-5_284_899_349_753} A white snooker ball of mass \(m\) moves with speed \(u\) towards a stationary black snooker ball of the same mass and radius. Taking the \(x\)-axis to be the line of centres of the two balls at the moment of collision, the direction of motion of the white ball before the collision makes an angle of \(30 ^ { \circ }\) with the positive \(x\)-axis (see diagram).

1. Given that the coefficient of restitution is 0.9 , find the angle made with the \(x\)-axis by the velocity of the white ball after the collision.
2. Show that after the collision the white ball cannot have a negative \(x\)-component of velocity whatever the value of the coefficient of restitution.

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.

11 A particle of mass 0.2 kg is projected so that it hits a smooth sloping plane \(\Pi\) that makes an angle of \(\sin ^ { - 1 } 0.6\) above the horizontal. The path of the particle is in a vertical plane containing a line of greatest slope of \(\Pi\). Immediately before the first impact between the particle and \(\Pi\), the particle is moving horizontally with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of restitution between the particle and \(\Pi\) is 0.5 .

1. Find the magnitude of the impulse on the particle from \(\Pi\) at the first impact, and state the direction of this impulse.
2. Find the distance between the points on \(\Pi\) where the first and second impacts occur.
3. Find the time taken between the first and third impacts.

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.

2 A smooth uniform ball travelling along a smooth horizontal table collides with a second smooth uniform ball of the same mass and radius which is at rest on the table. At the moment of impact the line of centres makes an angle of \(30 ^ { \circ }\) with the direction in which the first ball is moving. If the coefficient of restitution between the balls is \(e\), show that

1. the component of the first ball's velocity, along the line of centres, after the impact is $$\frac { \sqrt { 3 } u } { 4 } ( 1 - e )$$
2. the first ball is deflected by the impact through an angle \(\theta\), where $$\tan \theta = \frac { ( 1 + e ) \sqrt { 3 } } { 5 - 3 e }$$

\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]

Answer only one of the following two alternatives. EITHER A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of modulus of elasticity \(4mg\) and natural length \(l\). The other end of the string is attached to a fixed point \(O\). The particle rests in equilibrium at the point \(E\), vertically below \(O\). The particle is pulled down a vertical distance \(\frac{3l}{4}\) from \(E\) and released from rest. Show that the motion of \(P\) is simple harmonic with period \(\pi\sqrt{\left(\frac{l}{g}\right)}\). [4] At an instant when \(P\) is moving vertically downwards through \(E\), the string is cut. When \(P\) has descended a further distance \(\frac{3l}{4}\) under gravity, it strikes a fixed smooth plane which is inclined at 30° to the horizontal. The coefficient of restitution between \(P\) and the plane is \(\frac{1}{3}\). Show that the speed of \(P\) immediately after the impact is \(\frac{1}{3}\sqrt{(5gl)}\). [8] OR A new restaurant \(S\) has recently opened in a particular town. In order to investigate any effect of \(S\) on an existing restaurant \(R\), the daily takings, \(x\) and \(y\) in thousands of dollars, at \(R\) and \(S\) respectively are recorded for a random sample of 8 days during a six-month period. The results are shown in the following table.

1. Calculate the product moment correlation coefficient for this sample. [4]
2. Stating your hypotheses, test, at the 2.5% significance level, whether there is negative correlation between daily takings at the two restaurants and comment on your result in the context of the question. [5]

Another sample is taken over \(N\) randomly chosen days and the product moment correlation coefficient is found to be \(-0.431\). A test, at the 5% significance level, shows that there is evidence of negative correlation between daily takings in the two restaurants.

1. Find the range of possible values of \(N\). [3]

Answer only one of the following two alternatives. EITHER A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of modulus of elasticity \(4mg\) and natural length \(l\). The other end of the string is attached to a fixed point \(O\). The particle rests in equilibrium at the point \(E\), vertically below \(O\). The particle is pulled down a vertical distance \(\frac{3l}{4}\) from \(E\) and released from rest. Show that the motion of \(P\) is simple harmonic with period \(\pi\sqrt{\left(\frac{l}{g}\right)}\). [4] At an instant when \(P\) is moving vertically downwards through \(E\), the string is cut. When \(P\) has descended a further distance \(\frac{5l}{4}\) under gravity, it strikes a fixed smooth plane which is inclined at 30° to the horizontal. The coefficient of restitution between \(P\) and the plane is \(\frac{1}{3}\). Show that the speed of \(P\) immediately after the impact is \(\frac{1}{3}\sqrt{(5gl)}\). [8] OR A new restaurant \(S\) has recently opened in a particular town. In order to investigate any effect of \(S\) on an existing restaurant \(R\), the daily takings, \(x\) and \(y\) in thousands of dollars, at \(R\) and \(S\) respectively are recorded for a random sample of 8 days during a six-month period. The results are shown in the following table.

Day	1	2	3	4	5	6	7	8
\(x\)	1.2	1.4	0.9	1.1	0.8	1.0	0.6	1.5
\(y\)	0.3	0.4	0.6	0.6	0.25	0.75	0.6	0.35

1. Calculate the product moment correlation coefficient for this sample. [4]
2. Stating your hypotheses, test, at the 2.5\% significance level, whether there is negative correlation between daily takings at the two restaurants and comment on your result in the context of the question. [5]

Another sample is taken over \(N\) randomly chosen days and the product moment correlation coefficient is found to be \(-0.431\). A test, at the 5\% significance level, shows that there is evidence of negative correlation between daily takings in the two restaurants.

1. Find the range of possible values of \(N\). [3]

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 \(13.5 \leqslant \mu \leqslant 16.7\).
3. 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. $$12.2 \quad 13.3 \quad 16.4 \quad 14.0 \quad 13.9 \quad 15.4$$
4. 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]

Two smooth spheres \(A\) and \(B\), of equal radius, are moving in the same direction in the same straight line on a smooth horizontal table. Sphere \(A\) has mass \(m\) and speed \(u\) and sphere \(B\) has mass \(\alpha m\) and speed \(\frac{1}{4}u\). The spheres collide and \(A\) is brought to rest by the collision. Find the coefficient of restitution in terms of \(\alpha\). [6] Deduce that \(\alpha \geqslant 2\). [2]

Two smooth spheres \(A\) and \(B\), of equal radii and masses \(2m\) and \(m\) respectively, lie at rest on a smooth horizontal table. The spheres \(A\) and \(B\) are projected directly towards each other with speeds \(4u\) and \(3u\) respectively. The coefficient of restitution between the spheres is \(e\). Find the set of values of \(e\) for which the direction of motion of \(A\) is reversed in the collision. [5]