Teach A Level Maths Momentum and Collisions

Volume 4: Mechanics Momentum and Collisions

When a ball, A, on a pool table hits another, B, which is at rest, immediately after the impact the 2 balls move in many different ways. Two of the important things are the direction and speed of A at the impact. What quantities determine how the balls move immediately after the impact ? A B velocity

Two of the important things are the direction and speed of A at the impact. What quantities determine how the balls move immediately after the impact ? A B Another important factor is the angle between the line that A moves along and the line of the centres. velocity When a ball, A, on a pool table hits another, B, which is at rest, immediately after the impact the 2 balls move in many different ways.

Two of the important things are the direction and speed of A at the impact. What quantities determine how the balls move immediately after the impact ? A B Another important factor is the angle between the line that A moves along and the line of the centres. velocity When a ball, A, on a pool table hits another, B, which is at rest, immediately after the impact the 2 balls move in many different ways.

Two of the important things are the direction and speed of A at the impact. What quantities determine how the balls move immediately after the impact ? A B velocity In our modelling we will only consider the cases when A moves along the line of the centres. In this example, the masses of A and B are the same but in other impacts, the masses will be different. In those cases mass is also important. mass When a ball, A, on a pool table hits another, B, which is at rest, immediately after the impact the 2 balls move in many different ways.

e.g.1Find the momentum of a ball of mass 0·2 kg travelling at 3 m s -1. Newton gave the name momentum to the product of mass and velocity. momentum  mass  velocity Velocity is a vector, so momentum is a vector. momentum  0·2  3 Solution:  0·6 kg m s -1 The units of momentum are kg m s -1, the units of mass multiplied by velocity.

e.g.2 A and B are 2 particles. Find the momentum of each where, A has mass 4 kg and speed 3 m s B has mass 2 kg and speed of 5 m s -1 towards A AB Solution: Momentum of A  momentum  mass  velocity 4   12 kg m s -1 Since we are dealing with vectors, we must decide which direction is positive. Momentum of B  2  3 55  10 kg m s -1  Mass is written inside the circle. Velocity is shown by a number and an arrow.

When 2 bodies collide in a straight line, the motion of each immediately after the impact depends on the individual masses and velocities. e.g AB Before After AB ( Although A and B are drawn well apart, “before” means as they are about to collide and “after” means immediately after the collision. ) As a result of a collision, (a) one or both velocities can be reversed or (b) one body can be brought to rest or (c) the bodies can coalesce ( stick together ).

e.g.4 A is brought to rest. A and B coalesce AB Before After AB Directions of both are reversed AB Before AB After AB Before AB After (i) (ii) (iii) 6 1 A + B

Momentum AB Total Before After Before After Complete the momentum table for the diagrams below AB Before After AB (i) AB AB (ii)  (i) (ii) 12  10 44 6 8 4 Equal

The momentum of each body is changed by the collision BUT the total momentum remains the same. The “principle of the conservation of momentum” tells us that the total momentum before a collision = the total momentum after This equation lets us find the value of one unknown velocity or mass.

e.g.5A body, A, of mass 0·2 kg travelling with a speed of 5 m s -1 hits another body, B, moving in the same direction with speed 1 m s -1. The collision reduces the speed of A to 1 m s -1 and reverses its direction. Find the mass of B. Before After 0·2 5 A 1 B 2 B m m 1 A Always show a letter on the diagram for the unknown quantity. The speed of B increases to 2 m s -1 but its direction is unchanged.

e.g.5A body, A, of mass 0·2 kg travelling with a speed of 5 m s -1 hits another body, B, moving in the same direction with speed 1 m s -1. The collision reduces the speed of A to 1 m s -1 and reverses its direction. Before After 0·2 5 A 1 B 2 B m m 1 A The speed of B increases to 2 m s -1 but its direction is unchanged. No sign errors if we check the diagram for each term ! The sign between the terms is always  as we are adding the momentums. Solution: Conservation of Momentum Tot. mom. before  Tot. mom. after 0·2  5   m m  1  11  m m  2 1  m   0·2  2m2m  m1·2 The mass of B is 1·2 kg. Find m.

e.g.6Two particles, A and B, have mass 0·4 kg and 0·6 kg respectively. They are moving towards each other with speeds 1 m s -1 and 3 m s -1 respectively when they collide. Find the speed and direction of B after the collision. v Solution: Tot. mom. before  Tot. mom. after 0·4  1   0·6  33  0·4 22  0·6  v 0·4  1·8   0·8  0·6v  0·6v The speed of B is 1 m s -1 opposite to the direction shown in the diagram. Before 0·4 1 A 3 B 0·6 After 2 A 0·4 B 0·6  0·6  v   1 If you can’t spot which way B moves assume it goes in a +ve direction The collision increases the speed of A to 2 m s -1 and reverses its direction.

SUMMARY  Momentum  mass  velocity  Momentum is a vector.  The units of momentum are kg m s -1.  In a collision, the conservation of momentum gives: total momentum before  the total momentum after  Solving collision problems: Write the momentum statement (can abbreviate ). Write the equation remembering to add the momentums but watching for negative velocities. Draw before and after diagrams giving mass and velocity ( showing the direction with an arrow ). Draw to show the positive direction. 

Before 0·5 3 P 2 Q EXERCISE 1.Two particles P and Q are moving towards each other in a straight line with speeds 3 m s -1 and 2 m s -1. The mass of P is 0·5 kg. After QP The speed of Q is reduced to 1 m s -1. Copy and complete the diagram and find the mass of Q. The directions of both P and Q are reversed by the collision but the speed of P remains the same.

Before 0·5 3 P 2 Q EXERCISE Solution: After QP 1 m m 3 0·5 0·5  3  m m  22  33  m m  1 Conservation of Momentum Tot. mom. before  Tot. mom. after  3m3m 3  1·5  2m2m  1·5  m  m  1 The mass of Q is 1 kg.

EXERCISE 2.Two particles, A and B, of masses 0·6 kg and 1·4 kg respectively, are moving in the same direction in the same straight line. The speed of A is 4 m s -1 and of B is 1 m s -1. A catches up and collides with B and the particles coalesce. What is the combined speed after the impact ?

Before 0·6 4 A 1 B EXERCISE After v 1·4 2 0·6  4  1·4  1  2  v Conservation of Momentum Tot. mom. before  Tot. mom. after  3·8  2v2v The combined speed is 1·9 m s -1. A :mass 0·6 kg, speed 4 m s -1 B: mass 1·4 kg, speed 1 m s -1 v  1·9 Solution:

The following page contains the summary in a form suitable for photocopying.

 Momentum  mass  velocity  Momentum is a vector.  The units of momentum are kg m s -1.  In a collision, the conservation of momentum gives: total momentum before  the total momentum after  Solving collision problems: Write the momentum statement (can abbreviate ). Write the equation remembering to add the momentums but watching for negative velocities. Draw before and after diagrams giving mass and velocity ( showing the direction with an arrow ). Draw to show the positive direction.  Summary MOMENTUM AND COLLISIONS TEACH A LEVEL MATHS – MECHANICS 1