3.4 Momentum, Impulse & Collisions

Two objects  and  have masses 0.6 kg and  kg respectively. They are moving towards each other in opposite directions in a straight line on a smooth horizontal surface. The objects collide directly and coalesce to form an object . Immediately before the collision, is travelling with speed  and is travelling with speed  The speed of after the collision is

Find the value of .

There is a constant resistance force acting on  which causes to come to rest after travelling 120 m.

Find the magnitude of the resistance force acting on .

Two particles,  and , with masses  and  respectively are moving towards each other in opposite directions along the same straight line on a smooth horizontal plane when they collide directly. Immediately before the collision, the speed of is and the speed of is . Immediately after the collision, the speed of is  and the speed of  is .

The direction of motion of is reversed after the collision.

Show that .

State whether the direction of motion of  is reversed after the collision.

Given that  is double , find the values of  and .

Lucy and Fran are driving monster trucks along the same straight line in opposite directions towards each other. The combined mass of Lucy and her truck is 5400 kg and the combined mass of Fran and her truck is 4700 kg. They collide directly. Immediately before the collision, Lucy’s truck is moving with speed  and Fran’s truck is moving with speed . Immediately after the collision, Lucy and Fran’s monster trucks are moving at the same speed.

Given instead that Lucy and Fran’s monster trucks bounce off each other and move in opposite directions with equal speeds after the collision, calculate the speed at which they both travel.

A ball of mass 2 kg is projected along a rough horizontal floor in a straight line with speed  Initially the ball is 20 m away from the wall as shown in the diagram above. The coefficient of friction between the ball and the floor is 0.1.

Find the momentum of the ball immediately before it hits the wall.

The ball rebounds from the wall and travels for 3 s before coming to rest.

Find the magnitude of the impulse exerted on the wall by the ball.

Two particles  and , with masses 0.4 kg and 2.4 kg respectively, are moving in the same direction along a straight line on a smooth horizontal plane. At the instant when  has speed  and  has speed , the two particles collide directly. Immediately after the collision, the two particles are moving in opposite directions and the speed of  is half of the speed of .

Show that the speed of  immediately after the collision is 

After the collision continues travelling in the same straight line for 3 seconds before being brought to rest by a constant resistive force of magnitude 3.2 N.

Calculate the value of .

Paul, a teenager of mass , is skating in a straight line on a skateboard of mass . In the instant that Paul and the skateboard are travelling at , Paul jumps forward off the skateboard. Paul’s speed is  immediately after he jumps off the skateboard.

Two objects  and have masses 0.6 kg and 1.4 kg respectively. They are moving towards each other in opposite directions in a straight line on a smooth horizontal plane. The objects collide directly and coalesce to form an object . Immediately before the collision, is moving with speed 8 m s^-1 and is moving with speed 6 m s^-1.

By modelling the objects as particles, find the speed and direction of motion of  after the collision.

At the instance that object  is moving with speed , the magnitude of its momentum is 13 N s.

Find the value of .

Two marbles  and  have masses 3 kg and  kg respectively. The marbles are rolled towards each other in a straight line on a smooth horizontal table and they collide directly bringing  to rest. Immediately before the collision, the speeds of  and are  and  respectively. Immediately after the collision, moves with speed . The marbles are modelled as particles.

Find the value of .

Two particles,  and , with masses  and  respectively are moving towards each other in opposite directions along the same straight line on a smooth horizontal plane. They collide directly. Immediately before the collision, the speed of  is  and the speed of  is . Immediately after the collision, the speed of is halved and its direction of motion is reversed.

Find the speed of  immediately after the collision.

State whether the direction of motion of  is changed immediately after the collision.

Find the magnitude of the impulse exerted by  on  in the collision.

A car and its driver have a combined mass kg; it is travelling along a straight smooth horizontal road with constant speed 72 km h^-1. The driver applies a constant horizontal driving force,  N, for 6 seconds. When the driving force is removed, the car is moving at a constant speed 29 m s^-1. The magnitude of the impulse given to the car by the driving force is 11250 N s.

Find the value of .

Find the value of .

After driving at a constant speed 29 m s^-1, the driver applies a constant braking force 7250 N which causes the car to stop.

Two small balls and  have masses  kg and 1.5 kg respectively. They are moving in the same direction along the same straight line on a smooth horizontal table when they collide directly. Immediately before they collide, is moving with speed  and  is moving with speed . Immediately after they collide,  is moving with speed  and  is moving with speed . The direction of motion of is unchanged after the collision.

Find the value of  in the case that:

the direction of motion of  is unchanged after the collision.

the direction of motion of  is reversed after the collision.

Amber works at a circus and routinely performs the human cannonball act. The act involves Amber, mass 65 kg, being ejected horizontally out of a specially designed cannon, mass 300 kg, into a net 100 metres away. Both the cannon and Amber start from rest before Amber is shot out at a speed of 108 km h^-1.

Amber is travelling with a horizontal speed of 26 m s^-1 when she hits the net. The horizontal resistance force is assumed to be constant.

Find the time it takes Amber to reach the net after being ejected from the canon.

Calculate the magnitude of the resistance force.

Two particles  and  move towards each other in opposite directions along a straight line on a smooth horizontal surface. The mass of  is and the mass of is . They collide directly. Immediately before the collision,  moves with speed  and  moves with speed . Immediately after the collision the directions of  and  are reversed and  has speed  and  has speed .

Show that .

After the collision, both particles decelerate at a constant rate of 2 m s^-2. Particle takes  seconds longer to come to rest than particle .

Find the values of  and .

Harry drops a cricket ball, mass 0.16 kg, from a height 2 m above the ground. The only force acting on the cricket ball as it falls is the force due to gravity. The ball falls in a straight vertical line and hits the ground and bounces back upwards still following the same straight line. The magnitude of the impulse exerted on the ground by the ball is 1.44 kg m s^-1.

Find the greatest height above the ground reached by the cricket ball after it bounces off the ground.

An empty car of mass kg is stopped in the middle of a road. A lorry of mass 3800 kg is driving in at a constant speed of 48 km h^-1 when it collides with the car. The magnitude of the impulse exerted on the car by the lorry is 12500 N s.

Given that the speeds of the lorry and the car are equal immediately after the collision, find their common speed and find the value of .

In this question, take .

AB is a line of greatest slope, of length 4 m, on a smooth plane inclined at 40° to the horizontal. Particle  of mass 0.4 kg is projected down the slope from the point and particle  of mass 0.33 kg is projected up the slope from the point . When has speed  and is moving up the slope with speed , the particles collide directly at a point 1 m down the slope from , as shown in the diagram above. Immediately after the collision, the direction of motion of is reversed and it has speed .

Show that  reaches the top of the slope, , in the subsequent motion.

Determine, with calculations, whether  reaches before  reaches .

Armina and Dillon are two armadillos with masses 5 kg and  kg respectively. They curl into balls and roll towards each other in opposite directions along a straight line on a smooth ground. They collide directly which causes Armina’s direction of motion to be reversed. Immediately before the collision the speed of Armina is  and the speed of Dillon is . The magnitude of the impulse exerted on Dillon by Armina is 28 N s. Armina and Dillon are modelled as particles.

Find the speed of Armina immediately after she collides with Dillon.

Dillon’s speed is immediately after the collision.

Find the possible values of .

After the collision, Armina continues to roll with a constant speed until she reaches some rough ground. Once Armina enters the rough ground she continues to roll for 2 seconds before coming to rest.

Find the coefficient of friction between Armina and the rough ground.