Momentum and impulse

A football of mass 0.45 kg is kicked horizontally at a vertical wall. It hits the wall at right angles with speed 10 m s⁻¹ and rebounds along the same line with speed 6 m s⁻¹. The football is modelled as a particle.

Two trolleys run on a straight horizontal track, which is smooth. Trolley A, of mass 4 kg, moves at 3 m s⁻¹ and catches up with trolley B, of mass 2 kg, which is moving at 1.5 m s⁻¹ in the same direction. The trolleys collide and couple together. Both trolleys are modelled as particles.

A boat of mass 225 kg floats at rest on still water, with a man of mass 75 kg standing at rest in it. The man jumps horizontally from the boat, leaving it with speed 3 m s⁻¹. The resistance of the water is negligible, and the man and the boat are modelled as particles moving in the same straight line.

A particle P of mass 0.4 kg is moving on a smooth horizontal surface with velocity (5i + 2j) m s⁻¹, where i and j are perpendicular horizontal unit vectors. P receives an impulse of (−2.8i + 1.2j) N s.

A puck of mass 0.8 kg is at rest on smooth horizontal ice. From time t = 0 it is acted on by a horizontal force in a fixed line. The graph shows the component F newtons of this force, in a fixed positive direction along the line, against time t seconds. The force is constant within each interval shown, and is zero for t > 3. The puck is modelled as a particle.

Two particles A and B move along the same straight line on a smooth horizontal surface, and collide directly at t = 2. The graph shows the velocity v m s⁻¹ of each particle, in a fixed positive direction along the line, against time t seconds. The mass of A is 1.2 kg.

A ball of mass 0.25 kg moves on a smooth horizontal surface. It is struck, and its velocity changes from u to v. The figure shows u and v drawn to scale on a grid of squares of side 1 m s⁻¹, together with the perpendicular horizontal unit vectors i and j. The ball is modelled as a particle.

Two toy trucks P and Q are held at rest on a smooth straight horizontal track, with a compressed light spring between them. A buffer is fixed across the track at each end. The figure shows the masses of the trucks and the distance of each truck from the buffer it faces. The trucks are released, and the spring pushes them apart instantaneously. Immediately afterwards Q moves with speed 1.2 m s⁻¹. The trucks are modelled as particles.

Two particles A and B, of masses 2m and 3m, move towards each other along the same straight line on a smooth horizontal plane. Just before they collide, A has speed 4u and B has speed 2u. In the collision the direction of motion of A is reversed, and immediately afterwards A has speed u.

A particle A of mass 0.5 kg lies at rest on a smooth horizontal table. It is attached to one end of a light inextensible string, which passes over a small smooth pulley fixed at the edge of the table. The other end of the string is attached to a particle B of mass 0.3 kg. B is held at rest vertically below the pulley with the string slack, and is then released. B falls freely through 0.4 m, at which instant the string becomes taut. A does not reach the pulley in the motion considered.

A hammer head of mass 0.6 kg, moving horizontally at 9 m s⁻¹, strikes a nail of mass 0.015 kg that is at rest and pointing horizontally into a vertical wooden board. Immediately after the impact the hammer head and nail move together, and they come to rest after the nail has moved 0.025 m into the board. The board exerts a constant resistance on the nail, and all other horizontal forces are negligible. The hammer head and nail are modelled as particles.

A small box of mass 4 kg is at rest at the bottom of a rough plane inclined at an angle α to the horizontal, where tan α = ¾. The coefficient of friction between the box and the plane is 0.5. The box is given an impulse of magnitude 24 N s directed up a line of greatest slope of the plane. The box is modelled as a particle.

A firework of mass 0.9 kg is moving vertically upwards with speed 14 m s⁻¹ when it explodes into two parts, P of mass 0.3 kg and Q of mass 0.6 kg. Immediately after the explosion P moves vertically upwards with speed 35 m s⁻¹. The explosion is instantaneous, air resistance is negligible, and P and Q are modelled as particles.

Three particles A, B and C, of masses 2 kg, 1 kg and 3 kg, lie in a straight line, in that order, on a smooth horizontal surface. B and C are at rest. A is projected towards B with speed 6 m s⁻¹. A collides directly with B, and immediately afterwards A moves in its original direction with speed 2.5 m s⁻¹. B then collides with C, and B and C coalesce.

Two particles P and Q move along a straight line on a smooth horizontal surface. The line passes through a fixed point O. The graph shows the displacement s metres of each particle from O against time t seconds. The particles collide at t = 4 and coalesce, and the graph also shows the displacement of the combined particle up to t = 10. The mass of P is 3 kg.

A bullet is fired horizontally along a straight line on which two blocks A and B lie at rest on a smooth horizontal surface. The figure shows three stages of the motion, with the masses and the known speeds. In stage 2 the bullet has passed through A, and in stage 3 it has become embedded in B. The bullet passes through A in a negligible time and loses no mass. The bullet and blocks are modelled as particles moving in the same straight line.

A puck of mass 0.2 kg is at rest on a rough horizontal ice rink. At time t = 0 it is struck by a stick and slides in a straight line towards a vertical board. It hits the board at right angles and rebounds along the same line. The graph shows the velocity v m s⁻¹ of the puck, with the direction away from the stick as positive, against time t seconds, until the puck comes to rest. The contacts with the stick and with the board are modelled as instantaneous, and the puck is modelled as a particle.

A ball of mass 0.4 kg is released from rest and falls vertically onto horizontal ground. It bounces and rises vertically. The figure shows the height from which the ball is released and the greatest height it reaches after the bounce. Air resistance is negligible, and the ball is modelled as a particle.

A particle of mass 0.5 kg moves on a smooth horizontal plane. The unit vectors i and j are in the directions of the x- and y-axes shown in the figure, and distances are in metres. At time t = 0 the particle is at O. It moves with constant velocity to P, which it reaches at t = 2. At P it receives an impulse, and it then moves with a new constant velocity to Q, which it reaches at t = 4. The figure shows O, P and Q on a grid of 1 m squares, with t in seconds.

A truck moves along a straight horizontal railway track. A sack is released from rest from a hopper and falls vertically through the height shown in the figure onto the floor of the truck. The sack does not bounce, and immediately moves with the truck. The figure shows the masses and the speed of the truck just before the sack lands. The track is smooth, air resistance is negligible, the impact is instantaneous, and the truck and sack are modelled as particles. The unit vectors i and j are horizontal in the direction of motion of the truck and vertically upwards respectively.

Two particles A and B, of masses 0.4 kg and 0.6 kg, lie on a rough horizontal floor. B is at rest. A is projected along the floor directly towards B with speed 5 m s⁻¹ from a point 3 m from B. The coefficient of friction between each particle and the floor is μ. A collides directly with B. Immediately after the collision A is at rest. B then slides 1.5 m before coming to rest.

A pile driver of mass 400 kg is released from rest at a height of 1.8 m above the top of a pile of mass 600 kg, which is at rest with its lower end in the ground. The driver falls freely, strikes the pile and does not rebound, so that the driver and pile move downwards together. The ground exerts a constant resistance of 45 000 N on the pile while it moves. The driver and pile are modelled as particles.

Two particles A and B, of masses 3 kg and 2 kg, lie at rest on a rough horizontal floor, 1.2 m apart. They are connected by a light inextensible string of length 2 m, which is slack. The coefficient of friction between each particle and the floor is 0.25. A is given an impulse of magnitude 18 N s in the direction BA, directly away from B. When the string becomes taut, the particles move off together with the same speed.

A rifle of mass 3 kg fires a bullet of mass 0.02 kg horizontally with speed 400 m s⁻¹. The rifle is free to move horizontally and is initially at rest. The bullet travels horizontally, with no change in speed, and embeds itself in a block of wood of mass 1.98 kg, which is at rest on a rough horizontal table. The coefficient of friction between the block and the table is 0.4. The bullet and block are modelled as particles.

Two small blocks A and B are held at rest together at a point O on a rough horizontal floor, with a compressed light spring between them. The spring is released and pushes the blocks apart instantaneously along a straight line. Each block then slides until it comes to rest. The figure, drawn to scale, shows where A and B come to rest. The coefficient of friction between each block and the floor is 0.3, and the total mass of A and B is 5 kg. The blocks are modelled as particles.

The figure shows a particle P of mass 0.5 kg held at rest at a point A on a rough plane inclined at an angle α to the horizontal, where tan α = ¾. The plane meets a rough horizontal floor at the point F, and AF is a line of greatest slope. A particle Q of mass 0.9 kg is at rest on the floor at the point shown, with F and Q in the same vertical plane as A. The coefficient of friction between P and the plane is 0.25, and between each particle and the floor is 0.2. P is released, slides down the plane and passes onto the floor at F with no change in speed. It then slides along the floor, collides directly with Q, and the two particles coalesce.

Two particles A and B, of masses 2 kg and 1 kg, move towards each other along a straight horizontal track. The figure shows their positions and velocities at time t = 0, with distances x measured in metres from the fixed point O in the direction from A to B. The track is smooth for x < 10 and rough for x ≥ 10, and the coefficient of friction between each particle and the rough part of the track is 0.25. The particles collide directly, and immediately after the collision B moves in the positive x direction with speed 4 m s⁻¹. A and B are modelled as particles.

Two particles A and B, of masses 3 kg and 2 kg, are attached to the ends of a light inextensible string that passes over a small smooth pulley fixed above a horizontal floor, as shown in the figure. B rests on the floor, vertically below the pulley. A is held at rest, with the string slack, at the height above the floor shown. A is released and falls vertically. When A has fallen 0.9 m, the string becomes taut and B is jerked into motion. The parts of the string on each side of the pulley are vertical, and air resistance is negligible.