Collisions and restitution

A puck of mass 0.5 kg slides on smooth horizontal ice with velocity (3i + 4j) m s⁻¹. It is struck by a stick and receives an impulse of (2i − 6j) N s. The puck is modelled as a particle.

A model boat of mass 2 kg is drifting on a still pond with velocity (4i − j) m s⁻¹. A sudden gust of wind acts on it, and immediately afterwards its velocity is (−2i + 5j) m s⁻¹. The boat is modelled as a particle and the gust as an impulse; resistance from the water is ignored during the gust.

Two beads A and B, of masses 0.2 kg and 0.3 kg, are threaded on a smooth straight horizontal wire. A moves at 5 m s⁻¹ and B moves at 1 m s⁻¹ in the same direction, with A behind B, so that A catches B and they collide. Immediately after the collision, B moves at 3.4 m s⁻¹ in its original direction.

A small smooth sphere of mass 0.3 kg slides on a smooth horizontal floor with speed 4 m s⁻¹, moving at right angles towards a fixed smooth vertical wall. It rebounds from the wall along the same line with speed 2.8 m s⁻¹.

A small ball of mass 0.2 kg rolls on smooth horizontal ground, where i and j are perpendicular horizontal unit vectors. The ball is kicked. Figure 1 shows the velocity of the ball immediately before the kick, 10 m s⁻¹ at an angle θ above the i direction, and immediately after the kick, 13 m s⁻¹ at an angle α above the −i direction, where tan θ = ¾ and tan α = ¹²/₅. The ball is modelled as a particle.

Two gliders A and B move along a smooth straight horizontal air track. Figure 1 shows their masses and velocities immediately before and immediately after they collide. The gliders are modelled as particles.

A ball of mass 0.25 kg is released from rest above a horizontal floor and bounces. Figure 1 shows the velocity–time graph of the ball for the first 1.2 s, taking upwards as positive; labelled points give exact values. The ball is modelled as a particle moving vertically, with air resistance ignored. Take g = 9.8 m s⁻².

Two small lumps of putty A and B move towards each other along a straight line on a smooth horizontal floor, as shown in Figure 1. They collide and stick together to form a single lump C. Later, C strikes a fixed smooth vertical wall at right angles and rebounds along the same line with speed 1.5 m s⁻¹. The lumps are modelled as particles.

Two model railway trucks A and B, of masses 4 kg and 1 kg, run towards each other along a straight horizontal track. Just before they collide, A has speed 3 m s⁻¹ and B has speed 5 m s⁻¹. The coefficient of restitution between the trucks is 0.4. The trucks are modelled as particles and the track as smooth.

Two smooth spheres A and B, of equal radius and masses 3m and m, move towards each other along the same straight line on a smooth horizontal surface. Just before they collide, A has speed 2u and B has speed u. The coefficient of restitution between them is e.

Two pucks A and B slide on a smooth horizontal air table. A has mass 2 kg and velocity (3i + 2j) m s⁻¹; B has mass 3 kg and velocity (−i + 4j) m s⁻¹. The pucks collide and stick together. The pucks are modelled as particles.

Two smooth spheres P and Q, of equal radius and masses 0.5 kg and 1.5 kg, lie on a smooth horizontal floor. Q is at rest between P and a fixed smooth vertical wall, and the line of centres of P and Q is perpendicular to the wall. P is projected towards Q with speed 6 m s⁻¹. The coefficient of restitution between P and Q is 0.6, and between Q and the wall it is 0.4.

A tennis ball of mass 0.2 kg is released from rest at a height of 3.6 m above a horizontal concrete path. It bounces and first rises to a height of 1.6 m. The ball is modelled as a particle moving vertically, and air resistance is ignored.

Two small blocks A and B slide on smooth horizontal ice along the same straight line. A has mass 2 kg and moves at 5 m s⁻¹ towards B, which is at rest. After the collision A moves back along the line with speed 1 m s⁻¹ and B moves forward with speed 2 m s⁻¹. The blocks are modelled as particles.

Two particles A and B move along a straight line on a smooth horizontal surface and collide. Figure 1 shows the displacement–time graphs of A and B, where x metres is the displacement from a fixed point O on the line. The mass of A is 4 kg.

A ball of mass 0.5 kg moves on smooth horizontal ground. Figure 1 is a plan view, drawn on a 1 m grid, with i and j the unit vectors in the directions of the x- and y-axes. The ball passes through O at time t = 0 and moves with constant velocity to A, which it reaches when t = 2 s. At A it receives an impulse, and it then moves with constant velocity to B, which it reaches when t = 3.5 s. The ball is modelled as a particle.

A small smooth sphere of mass 0.4 kg moves on a smooth horizontal floor between two fixed smooth parallel vertical walls W1 and W2. The sphere moves along a line perpendicular to both walls. At time t = 0 it is at the point A shown in Figure 1, moving towards W2. The coefficient of restitution between the sphere and each wall is 0.5. The sphere is modelled as a particle.

Trolley A, of mass 1.5 kg, moves with speed 4 m s⁻¹ along a smooth straight horizontal track towards trolley B, of mass 2.5 kg, which is at rest. The trolleys collide. Figure 1 shows how the force F exerted on B by A varies with time during the collision, which lasts 40 ms. The trolleys are modelled as particles.

Figure 1 shows a smooth track in a vertical plane: two curved ramps joined by a straight horizontal section of floor. A small ball A, of mass 0.2 kg, is released from rest at the top of the left-hand ramp. A small ball B, of mass 0.3 kg, is at rest on the floor at the foot of the right-hand ramp. A moves down the ramp and along the floor and collides directly with B. The coefficient of restitution between A and B is 0.6. The balls are modelled as particles. Take g = 9.8 m s⁻².

A particle A, of mass 2 kg, moving with speed 3 m s⁻¹ on a smooth horizontal surface, collides directly with a particle B, of mass m kg, which is at rest. The coefficient of restitution between A and B is e. Figure 1 shows how the kinetic energy L joules lost in the collision depends on e.

Three small smooth spheres A, B and C, of equal radius and masses m, 2m and 4m, lie at rest in a straight line in that order on a smooth horizontal table. A is projected directly towards B with speed u. The coefficient of restitution between each pair of spheres is e.

A straight, narrow, smooth horizontal channel ends at a fixed vertical barrier that is perpendicular to the channel. Marble Q, of mass 0.2 kg, rests in the channel 2.8 m from the barrier. Marble P, of mass 0.3 kg and the same radius as Q, rolls along the channel towards Q and the barrier at 5 m s⁻¹. For impacts between the marbles e = 0.5; for an impact between Q and the barrier e = 0.8. Model the marbles as particles and ignore their rotation.

Two small smooth spheres A and B, of equal radius and masses m and 2m, move towards each other along the same straight line on a smooth horizontal floor, with speeds 2u and u respectively. The coefficient of restitution between the spheres is e, where 0 < e ≤ 1. A fixed smooth vertical wall stands perpendicular to the line of motion, on the far side of B from A.

A ball of mass 0.4 kg is rolling on smooth horizontal ground with velocity (12i + 5j) m s⁻¹, where i and j are perpendicular horizontal unit vectors. It receives an impulse of magnitude 5.2 N s in the direction of the vector (−5i + 12j). The ball is modelled as a particle.

Three small smooth spheres A, B and C, of equal radius, lie in a straight line on a smooth horizontal table. Their masses, the initial velocity of A and the distance between B and C are shown in Figure 1. B and C are at rest. The coefficient of restitution between any two of the spheres is 0.5. The spheres are modelled as particles.

Two particles A and B move along a straight line on a smooth horizontal floor towards a fixed smooth vertical wall which is perpendicular to the line. B is between A and the wall. A collides with B, and later B strikes the wall and rebounds. Figure 1 shows the velocity–time graphs of A and B for the first 3.2 s, with velocities towards the wall taken as positive. The mass of A is 2 kg.

A particle A, of mass 3 kg, moves with speed 4 m s⁻¹ on a smooth horizontal floor and collides directly with a particle B, of mass M kg, which is at rest. The coefficient of restitution between A and B is e. Figure 1 shows how the velocities of A and B immediately after the collision depend on e, with A's original direction of motion taken as positive.

Two particles P and Q, of masses 0.5 kg and 1 kg, move on a smooth horizontal plane, with i and j perpendicular horizontal unit vectors. P receives an impulse, which changes its velocity as shown in Figure 1. P then collides directly with Q, whose velocity is shown in the same figure; P and Q move along the same line just before the collision. The coefficient of restitution between P and Q is 0.5.