6.03k Newton's experimental law: direct impact

5. A particle \(A\) of mass \(3 m\) is moving in a straight line with speed \(2 u\) on a smooth horizontal floor. Particle \(A\) collides directly with another particle \(B\) of mass \(2 m\) which is moving along the same straight line with speed \(u\) but in the opposite direction to \(A\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 1 } { 3 }\).

1. 1. Show that the speed of \(B\) immediately after the collision is \(\frac { 7 } { 5 } u\)
   2. Find the speed of \(A\) immediately after the collision. After the collision, \(B\) hits a smooth vertical wall which is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the wall is \(\frac { 1 } { 2 }\). The first collision between \(A\) and \(B\) occurred at a distance \(x\) from the wall. The particles collide again at a distance \(y\) from the wall.
2. Find \(y\) in terms of \(x\).

1. A smooth sphere is moving with speed \(U\) in a straight line on a smooth horizontal plane. It strikes a fixed smooth vertical wall at right angles. The coefficient of restitution between the sphere and the wall is \(\frac { 1 } { 2 }\).

Find the fraction of the kinetic energy of the sphere that is lost as a result of the impact.
(5 marks)

5. A smooth sphere \(S\) of mass \(m\) is moving with speed \(u\) on a smooth horizontal plane. The sphere \(S\) collides with another smooth sphere \(T\), of equal radius to \(S\) but of mass \(k m\), moving in the same straight line and in the same direction with speed \(\lambda u , 0 < \lambda < \frac { 1 } { 2 }\). The coefficient of restitution between \(S\) and \(T\) is \(e\). Given that \(S\) is brought to rest by the impact,

1. show that \(e = \frac { 1 + k \lambda } { k ( 1 - \lambda ) }\).
2. Deduce that \(k > 1\).

8. A particle \(A\) of mass \(m\) is moving with speed \(3 u\) on a smooth horizontal table when it collides directly with a particle \(B\) of mass \(2 m\) which is moving in the opposite direction with speed \(u\). The direction of motion of \(A\) is reversed by the collision. The coefficient of restitution between \(A\) and \(B\) is \(e\).

1. Show that the speed of \(B\) immediately after the collision is \(\frac { 1 } { 3 } ( 1 + 4 e ) u\).
   (6)
2. Show that \(e > \frac { 1 } { 8 }\).
   (3) Subsequently \(B\) hits a wall fixed at right angles to the line of motion of \(A\) and \(B\). The coefficient of restitution between \(B\) and the wall is \(\frac { 1 } { 2 }\). After \(B\) rebounds from the wall, there is a further collision between \(A\) and \(B\).
3. Show that \(e < \frac { 1 } { 4 }\).
   (4) END

7. \begin{figure}[h] \end{figure} A light inextensible string of length \(a\) has one end attached to a fixed point \(O\) on a horizontal plane. A particle \(P\) is attached to the other end of the string. The particle is held at the point \(A\), where \(A\) is vertically above \(O\) and \(O A = a\). The particle is then projected horizontally with speed \(\sqrt { 10 a g }\), as shown in Figure 2. The particle strikes the plane at the point \(B\). After rebounding from the plane, \(P\) passes through \(A\). The coefficient of restitution between the plane and \(P\) is \(e\).

1. Show that \(e \geqslant \frac { 1 } { 2 }\) The point \(C\) is above the horizontal plane such that \(O C = a\) and angle \(C O B = 120 ^ { \circ }\) As the particle reaches \(C\), the string breaks. The particle now moves freely under gravity and strikes the plane at the point \(D\).
   Given that \(e = \frac { \sqrt { 3 } } { 2 }\)
2. find the size of the angle between the horizontal and the direction of motion of \(P\) at \(D\).

2. A particle \(P\) of mass 0.7 kg is moving in a straight line on a smooth horizontal surface. The particle \(P\) collides with a particle \(Q\) of mass 1.2 kg which is at rest on the surface. Immediately before the collision the speed of \(P\) is \(6 \mathrm {~ms} ^ { - 1 }\). Immediately after the collision both particles are moving in the same direction. The coefficient of restitution between the particles is \(e\).

1. Show that \(e < \frac { 7 } { 12 }\) Given that \(e = \frac { 1 } { 4 }\)
2. find the magnitude of the impulse exerted on \(Q\) in the collision.
   

5. Two small spheres \(A\) and \(B\) have mass \(3 m\) and \(2 m\) respectively. They are moving towards each other in opposite directions on a smooth horizontal plane, both with speed \(2 u\), when they collide directly. As a result of the collision, the direction of motion of \(B\) is reversed and its speed is unchanged.

1. Find the coefficient of restitution between the spheres. Subsequently, \(B\) collides directly with another small sphere \(C\) of mass \(5 m\) which is at rest. The coefficient of restitution between \(B\) and \(C\) is \(\frac { 3 } { 5 }\).
2. Show that, after \(B\) collides with \(C\), there will be no further collisions between the spheres.

7 \includegraphics[max width=\textwidth, alt={}, center]{f0813713-d677-4ed7-87e1-971a64bdb6ff-4_122_255_1503_561} The diagram shows two particles \(P\) and \(Q\), of masses 0.2 kg and 0.3 kg respectively, which move on a horizontal surface in the same direction along a straight line. A stationary particle \(R\) of mass 1.5 kg also lies on this line. \(P\) and \(Q\) collide and coalesce to form a combined particle \(C\). Immediately before this collision \(P\) has velocity \(4 \mathrm {~ms} ^ { - 1 }\) and \(Q\) has velocity \(2.5 \mathrm {~ms} ^ { - 1 }\).

1. Calculate the velocity of \(C\) immediately after this collision. At time \(t \mathrm {~s}\) after this collision the velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of \(C\) is given by \(v = V _ { 0 } - 3 t ^ { 2 }\) for \(0 < t \leqslant 0.3\). \(C\) strikes \(R\) when \(t = 0.3\).
2. (a) State the value of \(V _ { 0 }\).
   (b) Calculate the distance \(C\) moves before it strikes \(R\).
   (c) Find the acceleration of \(C\) immediately before it strikes \(R\). Immediately after \(C\) strikes \(R\), the particles have equal speeds but move in opposite directions.
3. Find the speed of \(C\) immediately after it strikes \(R\).

5 Two spheres \(A\) and \(B\), of equal radius, have masses \(m _ { 1 }\) and \(m _ { 2 }\) respectively. They lie at rest on a smooth horizontal plane. Sphere \(A\) is projected directly towards sphere \(B\) with speed \(u\) and, as a result of the collision, \(A\) is brought to rest. Show that

1. the speed of \(B\) immediately after the collision cannot exceed \(u\),
2. \(m _ { 1 } \leqslant m _ { 2 }\). \includegraphics[max width=\textwidth, alt={}, center]{15ed1dfc-8188-4e20-9c0b-ce31af35f0b6-3_273_611_1745_767} After the collision, \(B\) hits a smooth vertical wall which is at an angle of \(60 ^ { \circ }\) to the direction of motion of \(B\) (see diagram). In the impact with the wall \(B\) loses \(\frac { 2 } { 3 }\) of its kinetic energy. Find the coefficient of restitution between \(B\) and the wall and show that the direction of motion of \(B\) turns through \(90 ^ { \circ }\).

5 \includegraphics[max width=\textwidth, alt={}, center]{f6887893-66c5-40df-ba8d-9439a5c268eb-3_531_908_1674_616} A rectangular pool table \(K L M N\) has \(K L = a\) and \(K N = 2 a\). A ball lies at rest on the table just outside the pocket at \(L\) and is projected along the table with speed \(u\) in a direction making an angle \(\theta\) with the edge \(L M\). The ball hits the edge \(K N\) at \(Y\), rebounds to hit the edge \(L M\) at \(X\) and then rebounds into the pocket at \(N\). Angle \(L X Y\) is denoted by \(\phi\) (see diagram). The coefficient of restitution between the ball and an edge is \(\frac { 3 } { 4 }\), and all resistances to motion may be neglected. Show that \(\tan \phi = \frac { 3 } { 4 } \tan \theta\). [3] Show that \(X M = \left( 2 - \frac { 7 } { 3 } \cot \theta \right) a\), and find the value of \(\theta\). Find the speed with which the ball reaches \(N\), giving the answer in the form \(k u\), where \(k\) is correct to 3 significant figures.

4 Two uniform spheres \(A\) and \(B\), of equal radius, are at rest on a smooth horizontal table. Sphere \(A\) has mass \(3 m\) and sphere \(B\) has mass \(m\). Sphere \(A\) is projected directly towards \(B\), with speed \(u\). The coefficient of restitution between the spheres is 0.6 . Find the speeds of \(A\) and \(B\) after they collide. Sphere \(B\) now strikes a wall that is perpendicular to its path, rebounds and collides with \(A\) again. The coefficient of restitution between \(B\) and the wall is \(e\). Given that the second collision between \(A\) and \(B\) brings \(A\) to rest, find \(e\).

1 Three small spheres, \(A , B\) and \(C\), of masses \(m , k m\) and \(6 m\) respectively, have the same radius. They are at rest on a smooth horizontal surface, in a straight line with \(B\) between \(A\) and \(C\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 1 } { 2 }\) and the coefficient of restitution between \(B\) and \(C\) is \(e\). Sphere \(A\) is projected towards \(B\) with speed \(u\) and is brought to rest by the subsequent collision. Show that \(k = 2\). Given that there are no further collisions after \(B\) has collided with \(C\), show that \(e \leqslant \frac { 1 } { 3 }\).

3 Two uniform small smooth spheres \(A\) and \(B\), of masses \(m\) and \(2 m\) respectively, and with equal radii, are at rest on a smooth horizontal surface. Sphere \(A\) is projected directly towards \(B\) with speed \(u\), and collides with \(B\). After this collision, sphere \(B\) collides directly with a fixed smooth vertical barrier. The total kinetic energy of the spheres after this second collision is equal to one-ninth of its value before the first collision. Given that the coefficient of restitution between \(B\) and the barrier is 0.5 , find the coefficient of restitution between \(A\) and \(B\).

3 Three small smooth spheres \(A , B\) and \(C\) have equal radii and have masses \(m , 9 m\) and \(k m\) respectively. They are at rest on a smooth horizontal table and lie in a straight line with \(B\) between \(A\) and \(C\). The coefficient of restitution between any pair of the spheres is \(e\). Sphere \(A\) is projected directly towards \(B\) with speed \(u\). Given that half of the total kinetic energy is lost as result of the collision between \(A\) and \(B\), find the value of \(e\). After \(B\) and \(C\) collide they move in the same direction and the speed of \(C\) is twice the speed of \(B\). Find the value of \(k\).

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 \(e ^ { \prime }\).

2 Two uniform small spheres \(A\) and \(B\) have equal radii and masses \(4 m\) 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. Show that after the collision \(A\) moves with speed \(\frac { 1 } { 5 } u ( 4 - e )\) and find the speed of \(B\).
   Sphere \(B\) continues to move until it collides with 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 \(\frac { 3 } { 4 } e\). After this collision, the speeds of \(A\) and \(B\) are equal.
2. Find the value of \(e\).
   The spheres \(A\) and \(B\) now collide directly again.
3. Determine whether sphere \(B\) collides with the barrier for a second time.

3 Three uniform small spheres \(A , B\) and \(C\) have equal radii and masses \(2 m , 4 m\) 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 \(2 u\) and \(u\) respectively. The first collision is between \(A\) and \(B\).

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

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 \(A O B\) is a horizontal diameter. Initially, \(P\) is projected vertically downwards with speed \(\sqrt { } \left( \frac { 21 } { 2 } a g \right)\) 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 \(4 m\), 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.
   In its subsequent motion, \(P\) loses contact with the inner surface of the sphere at the point \(D\), where the angle between \(O D\) and the upward vertical through \(O\) is \(\theta\).
2. Find \(\cos \theta\).

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 Two smooth spheres \(A\) and \(B\), of equal radii, have masses 0.1 kg and \(m \mathrm {~kg}\) respectively. They are moving towards each other in a straight line on a smooth horizontal table and collide directly. Immediately before collision the speed of \(A\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Assume that in the collision \(A\) does not change direction. The speeds of \(A\) and \(B\) after the collision are \(v _ { A } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(v _ { B } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. Express \(m\) in terms of \(v _ { A }\) and \(v _ { B }\), and hence show that \(m < 0.25\).
2. Assume instead that \(m = 0.2\) and that the coefficient of restitution between the spheres is \(\frac { 1 } { 2 }\). Find the magnitude of the impulse acting on \(A\) in the collision.

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.

5 Two uniform small smooth spheres \(A\) and \(B\), of equal radii, have masses \(2 m\) and \(m\) respectively. They lie at rest on a smooth horizontal plane. Sphere \(A\) is projected directly towards \(B\) with speed \(u\). After the collision \(B\) goes on to collide directly with a fixed smooth vertical barrier, before colliding with \(A\) again. The coefficient of restitution between \(A\) and \(B\) is \(\frac { 2 } { 3 }\) and the coefficient of restitution between \(B\) and the barrier is \(e\). After the second collision between \(A\) and \(B\), the speed of \(B\) is five times the speed of \(A\). Find the two possible values of \(e\).

A smooth sphere, with centre \(O\) and radius \(a\), is fixed on a smooth horizontal plane \(\Pi\). A particle \(P\) of mass \(m\) is projected horizontally from the highest point of the sphere with speed \(\sqrt { } \left( \frac { 2 } { 5 } g a \right)\). While \(P\) remains in contact with the sphere, the angle between \(O P\) and the upward vertical is denoted by \(\theta\). Show that \(P\) loses contact with the sphere when \(\cos \theta = \frac { 4 } { 5 }\). Subsequently the particle collides with the plane \(\Pi\). The coefficient of restitution between \(P\) and \(\Pi\) is \(\frac { 5 } { 9 }\). Find the vertical height of \(P\) above \(\Pi\) when the vertical component of the velocity of \(P\) first becomes zero.

2 \includegraphics[max width=\textwidth, alt={}, center]{2c6b6722-ebba-4ade-9a9d-dd70e61cf52b-2_312_409_525_868} A small smooth ball \(P\) is moving on a smooth horizontal plane with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It strikes a smooth vertical barrier at an angle \(\alpha\) (see diagram). The coefficient of restitution between \(P\) and the barrier is 0.4 . Given that the speed of \(P\) is halved as a result of the collision, find the value of \(\alpha\).

2 \includegraphics[max width=\textwidth, alt={}, center]{5d40f5b4-e3d4-482c-8d8d-05a01bd3b43f-2_312_409_525_868} A small smooth ball \(P\) is moving on a smooth horizontal plane with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It strikes a smooth vertical barrier at an angle \(\alpha\) (see diagram). The coefficient of restitution between \(P\) and the barrier is 0.4 . Given that the speed of \(P\) is halved as a result of the collision, find the value of \(\alpha\).