6.03k Newton's experimental law: direct impact

\includegraphics{figure_2} A small smooth ball \(P\) is moving on a smooth horizontal plane with speed \(4\text{ m s}^{-1}\). 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 \(\alpha\). [5]

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]

Two uniform small smooth spheres \(A\) and \(B\) have equal radii and masses \(5m\) and \(2m\) 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 \(2u\). The coefficient of restitution between the spheres is \(e\).

1. Show that the speed of \(B\) after the collision is \(\frac{1}{7}u(1 + 15e)\) and find an expression for the speed of \(A\). [4]

In the collision, the speed of \(A\) is halved and its direction of motion is reversed.

1. Find the value of \(e\). [2]
2. For this collision, find the ratio of the loss of kinetic energy of \(A\) to the loss of kinetic energy of \(B\). [3]

Two uniform small smooth spheres \(A\) and \(B\) have equal radii and masses \(2m\) 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 \(\frac{2}{3}\).

1. Find, in terms of \(u\), the speeds of \(A\) and \(B\) after this collision. [4]
2. 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 \(\frac{1}{2}\). Find, in terms of \(d\) and \(u\), the time that elapses between the first and second collisions between \(A\) and \(B\). [5]

Three uniform small spheres \(A\), \(B\) and \(C\) have equal radii and masses \(5m\), \(5m\) and \(3m\) respectively. The spheres are at rest on a smooth horizontal surface, in a straight line, 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. Show that the speed of \(A\) after its collision with \(B\) is \(\frac{1}{2}u(1 - e)\) and find the speed of \(B\). [3]

Sphere \(B\) now collides with sphere \(C\). Subsequently there are no further collisions between any of the spheres.

1. Find the set of possible values of \(e\). [6]

Two particles \(A\) and \(B\), of masses 0.4 kg and 0.2 kg respectively, are moving down the same line of greatest slope of a smooth plane. The plane is inclined at 30° to the horizontal, and \(A\) is higher up the plane than \(B\). When the particles collide, the speeds of \(A\) and \(B\) are 3 m s\(^{-1}\) and 2 m s\(^{-1}\) respectively. In the collision between the particles, the speed of \(A\) is reduced to 2.5 m s\(^{-1}\).

1. Find the speed of \(B\) immediately after the collision. [2]

After the collision, when \(B\) has moved 1.6 m down the plane from the point of collision, it hits a barrier and returns back up the same line of greatest slope. \(B\) hits the barrier 0.4 s after the collision, and when it hits the barrier, its speed is reduced by 90%. The two particles collide again 0.44 s after their previous collision, and they then coalesce on impact.

1. Show that the speed of \(B\) immediately after it hits the barrier is 0.5 m s\(^{-1}\). Hence find the speed of the combined particle immediately after the second collision between \(A\) and \(B\). [7]

Two particles \(A\) and \(B\) have masses \(3m\) and \(2m\) respectively. Initially \(A\) is at rest and \(B\) is moving with speed \(u\) in a straight line towards \(A\). The coefficient of restitution between the particles is \(e\).

1. Find the speeds of the particles immediately after the collision.
2. Find the condition on \(e\) for \(A\) to be moving faster than \(B\) after the collision.

[8]

A particle \(P\) of mass \(m\) is moving with speed \(u\) on a fixed smooth horizontal surface. The particle strikes a fixed vertical barrier. At the instant of impact the direction of motion of \(P\) makes an angle \(\alpha\) with the barrier. The coefficient of restitution between \(P\) and the barrier is \(e\). As a result of the impact, the direction of motion of \(P\) is turned through \(90°\).

1. Show that \(\tan^2 \alpha = \frac{1}{e}\). [3]

The particle \(P\) loses two-thirds of its kinetic energy in the impact.

1. Find the value of \(\alpha\) and the value of \(e\). [5]

\includegraphics{figure_5} Two uniform smooth spheres \(A\) and \(B\) of equal radii each have mass \(m\). The two spheres are each moving with speed \(u\) on a horizontal surface when they collide. Immediately before the collision \(A\)'s direction of motion makes an angle of \(\alpha°\) with the line of centres, and \(B\)'s direction of motion is perpendicular to that of \(A\) (see diagram). The coefficient of restitution between the spheres is \(e\). Immediately after the collision, \(B\) moves in a direction at right angles to the line of centres.

1. Show that \(\tan \alpha = \frac{1+e}{1-e}\). [4]

\includegraphics{figure_6} Two uniform smooth spheres A and B of equal radii each have mass \(m\). The two spheres are each moving with speed \(u\) on a horizontal surface when they collide. Immediately before the collision, A's direction of motion makes an angle \(\alpha\) with the line of centres, and B's direction of motion makes an angle \(\beta\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac{1}{3}\) and \(2\cos\beta = \cos\alpha\).

1. Show that the direction of motion of A after the collision is perpendicular to the line of centres. [4]

The total kinetic energy of the spheres after the collision is \(\frac{3}{4}mu^2\).

1. Find the value of \(\alpha\). [4]

\includegraphics{figure_6} Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(km\) respectively. Sphere \(A\) is moving with speed \(u\) on a smooth horizontal surface when it collides with sphere \(B\) which is at rest. Immediately before the collision, \(A\)'s direction of motion makes an angle \(\theta\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac{1}{3}\).

1. Show that the speed of \(B\) after the collision is \(\frac{4u \cos \theta}{3(1 + k)}\). [3]

70% of the total kinetic energy of the spheres is lost as a result of the collision.

1. Given that \(\tan \theta = \frac{1}{3}\), find the value of \(k\). [6]

Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(km\) respectively. The two spheres are on a horizontal surface. Sphere \(A\) is travelling with speed \(u\) towards sphere \(B\) which is at rest. The spheres collide. Immediately before the collision, the direction of motion of \(A\) makes an angle \(\alpha\) with the line of centres. The coefficient of restitution between the spheres is \(\frac{1}{2}\).

1. Show that the speed of \(B\) after the collision is \(\frac{3u \cos \alpha}{2(1 + k)}\) and find also an expression for the speed of \(A\) along the line of centres after the collision, in terms of \(k\), \(u\) and \(\alpha\). [4]

After the collision, the kinetic energy of \(A\) is equal to the kinetic energy of \(B\).

1. Given that \(\tan \alpha = \frac{2}{3}\), find the possible values of \(k\). [5]

\includegraphics{figure_2} A particle \(P\) of mass \(m\) is moving with speed \(u\) on a fixed smooth horizontal surface. It collides at an angle \(\alpha\) with a fixed smooth vertical barrier. After the collision, \(P\) moves at an angle \(\theta\) with the barrier, where \(\tan\theta = \frac{1}{3}\) (see diagram). The coefficient of restitution between \(P\) and the barrier is \(e\). The particle \(P\) loses 20% of its kinetic energy as a result of the collision. Find the value of \(e\). [5]

\includegraphics{figure_4} Two identical smooth uniform spheres \(A\) and \(B\) each have mass \(m\). The two spheres are moving on a smooth horizontal surface when they collide with speeds \(u\) and \(2u\) respectively. Immediately before the collision, \(A\)'s direction of motion makes an angle of \(30°\) with the line of centres, and \(B\)'s direction of motion is perpendicular to the line of centres (see diagram). After the collision, \(A\) and \(B\) are moving in the same direction. The coefficient of restitution between the spheres is \(e\).

1. Find the value of \(e\). [5]
2. Find the loss in the total kinetic energy of the spheres as a result of the collision. [3]

Two smooth uniform spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(5m\) respectively. Sphere \(A\) is moving on a smooth horizontal surface with speed \(u\) when it collides with sphere \(B\) which is at rest on the surface. Immediately before the collision, \(A\)'s direction of motion makes an angle of \(\theta\) with the line of centres. After the collision, the kinetic energies of \(A\) and \(B\) are equal. The coefficient of restitution between the spheres is \(\frac{1}{3}\). \includegraphics{figure_1} Find the value of \(\tan\theta\). [6]

\includegraphics{figure_1} Two smooth uniform spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(5m\) respectively. Sphere \(A\) is moving on a smooth horizontal surface with speed \(u\) when it collides with sphere \(B\) which is at rest on the surface. Immediately before the collision, \(A\)'s direction of motion makes an angle of \(\theta\) with the line of centres. After the collision, the kinetic energies of \(A\) and \(B\) are equal. The coefficient of restitution between the spheres is \(\frac{1}{3}\). Find the value of \(\tan\theta\). [6]

\includegraphics{figure_1} Two smooth uniform spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(2m\) respectively. The two spheres are moving on a smooth horizontal surface when they collide with speeds \(u\) and \(\frac{1}{2}u\) respectively. Immediately before the collision, \(A\)'s direction of motion is along the line of centres, and \(B\)'s direction of motion makes an angle \(\theta\) with the line of centres (see diagram). As a result of the collision, the direction of motion of \(A\) is reversed and its speed is reduced to \(\frac{1}{4}u\). The direction of motion of \(B\) again makes an angle \(\theta\) with the line of centres, but on the opposite side of the line of centres. The speed of \(B\) is unchanged. Find the value of the coefficient of restitution between the spheres. [4]

Two smooth spheres \(A\) and \(B\) have equal radii and masses \(m\) and \(2m\) respectively. Sphere \(B\) is at rest on a smooth horizontal floor. Sphere \(A\) is moving on the floor with velocity \(u\) and collides directly with \(B\). The coefficient of restitution between the spheres is \(e\).

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

Subsequently, \(B\) collides with a fixed vertical wall which makes an angle \(\theta\) with the direction of motion of \(B\), where \(\tan\theta = \frac{3}{4}\). The coefficient of restitution between \(B\) and the wall is \(\frac{2}{3}\). Immediately after \(B\) collides with the wall, the kinetic energy of \(B\) is \(\frac{5}{27}\) of the kinetic energy of \(B\).

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

\includegraphics{figure_7} The smooth vertical walls \(AB\) and \(CB\) are at right angles to each other. A particle \(P\) is moving with speed \(u\) on a smooth horizontal floor and strikes the wall \(CB\) at an angle \(\alpha\). It rebounds at an angle \(\beta\) to the wall \(CB\). The particle then strikes the wall \(AB\) and rebounds at an angle \(\gamma\) to that wall (see diagram). The coefficient of restitution between each wall and \(P\) is \(e\).

1. Show that \(\tan \beta = e \tan \alpha\). [3]
2. Express \(\gamma\) in terms of \(\alpha\) and explain what this result means about the final direction of motion of \(P\). [4]

As a result of the two impacts the particle loses \(\frac{8}{9}\) of its initial kinetic energy.

1. Given that \(\alpha + \beta = 90°\), find the value of \(e\) and the value of \(\tan \alpha\). [4]

\includegraphics{figure_5} Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(\frac{2}{3}m\) respectively. The two spheres are each moving with speed \(u\) on a horizontal surface when they collide. Immediately before the collision \(A\)'s direction of motion is along the line of centres, and \(B\)'s direction of motion makes an angle of \(60°\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac{2}{3}\).

1. Find the angle through which the direction of motion of \(B\) is deflected by the collision. [6]
2. Find the loss in the total kinetic energy of the system as a result of the collision. [3]

\includegraphics{figure_7} Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(\frac{1}{2}m\) respectively. The two spheres are moving on a horizontal surface when they collide. Immediately before the collision, sphere \(A\) is travelling with speed \(u\) and its direction of motion makes an angle \(\alpha\) with the line of centres. Sphere \(B\) is travelling with speed \(2u\) and its direction of motion makes an angle \(\beta\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac{5}{8}\) and \(\alpha + \beta = 90°\).

1. Find the component of the velocity of \(B\) parallel to the line of centres after the collision, giving your answer in terms of \(u\) and \(\alpha\). [4]

The direction of motion of \(B\) after the collision is parallel to the direction of motion of \(A\) before the collision.

1. Find the value of \(\tan \alpha\). [5]

\includegraphics{figure_1} Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(2m\) respectively. The two spheres are moving with equal speeds \(u\) on a smooth horizontal surface when they collide. Immediately before the collision, \(A\)'s direction of motion makes an angle of \(60°\) with the line of centres, and \(B\)'s direction of motion makes an angle \(\theta\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(e\). After the collision, the component of the velocity of \(A\) along the line of centres is \(v\) and \(B\) moves perpendicular to the line of centres. Sphere \(A\) now has twice as much kinetic energy as sphere \(B\).

1. Show that \(v = \frac{1}{2}u(4\cos\theta - 1)\). [1]
2. Find the value of \(\cos\theta\). [4]
3. Find the value of \(e\). [2]

\includegraphics{figure_4} Two smooth vertical walls meet at right angles. The smooth sphere \(A\), with mass \(m\), is at rest on a smooth horizontal surface and is at a distance \(d\) from each wall. An identical smooth sphere \(B\) is moving on the horizontal surface with speed \(u\) at an angle \(\theta\) with the line of centres when the spheres collide (see diagram). After the collision, the spheres take the same time to reach a wall. The coefficient of restitution between the spheres is \(\frac{1}{2}\).

1. Find the value of \(\tan \theta\). [4]
2. Find the percentage loss in the total kinetic energy of the spheres as a result of this collision. [3]