5.1 Momentum & Collisions

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  before coming to rest.

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

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.

Three small balls  and  of equal radius have masses   and  respectively. The three balls lie in the same straight line on a smooth horizontal ground with in the middle of  and . and  are projected towards each other in opposite directions and collide directly when they are travelling with speeds  and  respectively. Immediately after the collision, the speed of  is twice the speed of

Find the two possible values for the speed of  immediately after the collision.

Immediately after the collision the speed of  is the larger of the two values found in part (a). In the subsequent motion,  collides with  which is at rest. After the collision,  moves with speed 

Given that there are no further collisions between  and , find the maximum 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.

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 spheres  and  are of equal radius and have masses  and  respectively.  and  are moving in the same direction in a straight line on a smooth horizontal surface when they collide directly. Immediately before the collision,  and  are moving with speeds  and  respectively. Immediately after the collision, the speed of  is , the speed of  is  and the direction of motion of  is unchanged.

Find the two possible values of .

Show that the minimum total kinetic energy lost in the collision is 

Three uniform spheres  and  of equal radius move in the same direction on the same straight line on a smooth horizontal table with  in the middle of  and . The masses of  and  are and respectively and  moves with speed  while  and  are at rest.  and  collide directly which does not change the direction of motion of  and subsequently  moves with speed 

Show that the speed of  immediately after it collides with is

In the subsequent motion,  collides directly with . Immediately after this collision,  moves with speed 

Show that there will be another collision between  and .

Given that A and B coalesce in this collision,

show that the speed of the combined spheres is 

show that the total loss of kinetic energy due to the three collisions is  correct to 3 significant figures.

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

Show that .

After the collision, both particles decelerate at a constant rate of  . Particle P takes  seconds longer to come to rest than particle Q.

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, hits the ground and bounces back upwards still following the same straight line. Immediately after the ball hits the ground, the magnitude of its momentum decreases by .

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 at a constant speed of  when it collides with the car. The magnitude of the momentum of the car immediately after the collision is .

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  to the horizontal. Particle P of mass 0.4 kg is projected down the slope from the point A and particle Q of mass 0.33 kg is projected up the slope from the point B. When P has speed  and Q is moving up the slope with speed , the particles collide directly at a point 1 m down the slope from A, as shown in the diagram above. Immediately after the collision, the direction of motion of Q is reversed and it has speed .

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

Determine, with calculations, whether P reaches A before Q reaches B.

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 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 . Immediately after the collision, the speed of Armina is  and the speed of Dillon is . Armina and Dillon are modelled as particles.

Find the two possible values for the loss of kinetic energy in the collision.

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.

A particle P of mass 5 kg is travelling along a straight line on a smooth horizontal surface with constant speed . It collides directly with particle Q of mass 2 kg which is moving along the same straight line. Immediately after the collision, the direction of motion of P is reversed and its speed is . The kinetic energy lost in the collision is 60 J.

R, S and T are three small marbles of equal radius and masses 0.4 kg, 0.2 kg and 0.1 kg respectively. The three marbles lie in the same straight line on a smooth horizontal floor with S in the middle. R and S are projected towards each other with speeds  and  respectively and collide directly. At the instant that R and S collide, T is projected towards S with a speed of . In the subsequent motion, R travels towards T with a speed of  and S travels towards T with a speed of . S and T collide directly and T rebounds with speed 

Given that there are no further collisions, find the maximum value of .