Teach A LevelTeach A Level MathsMaths

MomentMomentumumand and

CollisionsCollisions

Volume 4: Mechanics Volume 4: Mechanics Momentum and Momentum and

CollisionsCollisions

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.1 Find the momentum of a ball of mass 0·2 kg travelling at 3m 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 -

1 . . .B has mass 2 kg and speed of 5 m s

-1 towards A.

4 23 5A B

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

510 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.3

4 23 1A B

Before

After 4 21 3A B

( 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.

4 23 5A B

Before

After 4 21 3A B

Directions of both are reversed.

4 21 1A B

Before

4 20 1A B

After

4 22 1A B

Before

4 21A B

After

(i)

(ii)

(iii)

61

A + B

Momentum A B Total

Before

After

Before

After

Complete the momentum table for the diagrams below.

4 23 5A B

BeforeAfter 4 2

1 3A B

(i)4 2

2 2A B

4 20 2A B

(ii)

(i)

(ii)

12 10

4 6

8 40 4

2

2

4

4

Equal

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

afterThis equation lets us find the value of one unknown velocity or mass.

e.g.5 A 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.

BeforeAfter

0·25A 1B

2B

m

m1 A

0·2

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.5 A 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.

BeforeAfter

0·25A 1B

2B

m

m1 A

0·2The 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. after0·2

5 m 1 0·2 1 m 21 m 0·2 2m

m1·2The mass of B is 1·2 kg.

Find m.

e.g.6 Two 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. after0·4

1 0·6 3 0·4 2 0·6

v0·4 1·8 0·8 0·6v

0·6vThe speed of B is 1 m s

-1 opposite to the direction shown in the diagram.

Before 0·4

1A 3 B0·6

After 2 A0·4

B0·6

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

3P 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

3P 2 Q

EXERCISESolution

:

After QP 1

m

m3

0·5

0·5

3 m 2 0·5 3 m 1

Conservation of MomentumTot. mom. before Tot. mom.

after

3m3 1·5 2m 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

0·64A 1B

EXERCISE

After v

1·4

2

0·6

4 1·4

1 2 v

Conservation of MomentumTot. mom. before Tot. mom.

after 3·8 2v

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

v1·9

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