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What is momentum?

If mass is measured in kg and velocity in m s–1, what are the units of momentum?

p = m v

momentum = mass × velocity

The units are kg m s–1. (This can also be expressed as N s).

Momentum is a property of objects with mass and velocity.

It is a vector quantity with the same direction as the velocity of the object.

What is the equation for momentum?


Momentum as a vector


Force and momentum

In a tennis match, when a player exerts a force on the ball, it changes momentum. This means the ball can change speed, direction, shape or size, etc.

In order to change the momentum of an object, a force must be applied (from Newton’s first law).

The rate of change of momentum of an object is proportional to the resultant force acting on the object.

This is an alternative way of stating Newton’s second law in terms of momentum.


Rate of change of momentum

Stated mathematically, Newton’s second law is:F =

Δ(mv)Δt

For constant masses this becomes F =m Δv

Δt= m a

It can therefore be seen that the familiar equation F = m a is a special case of the more general equation for Newton’s second law in terms of momentum.

The more general form of the equation is necessary when mass is not constant, for example for a space shuttle taking off. The mass decreases as fuel is burned.


True or false?


Using the momentum equations


Conservation of linear momentum

The principle of conservation of linear momentum states:

The total linear momentum of a system of interacting bodies is constant, providing no external forces act.

This applies to collisions, where objects move together and hit one other, and to explosions, where objects fly apart from one another after initially being at rest.

collision explosion


Collisions


Explosions


Calculations: conservation of momentum


Elastic and inelastic collisions


Momentum and energy

When one ball is swung on Newton’s cradle, one ball moves out at the other end. If two balls are swung, two balls move out.

Total energy is also conserved in each collision because energy cannot be created or destroyed.

Therefore, why do the balls eventually come to rest?

Because kinetic energy is not always conserved, but is converted to other forms, in this case sound and heat. The collisions are therefore inelastic.

Two balls have double the mass, and double the momentum. In each collision, momentum is conserved.


Equation for kinetic energy

An equation for kinetic energy, Ek, in terms of momentum can be derived for a non-relativistic particle.

Ek =m v2

2

p = m v therefore v =pm

Substituting in for v in the equation for Ek:

( )Ek =

m m2

2

p

=m p2

2 m2

p2

2 mEk =


Fill in the missing words


What is impulse?

Newton’s second law in the form F = Δ(mv) / Δt can be rearranged to give: F Δt = Δ(mv)

The quantity F Δt is the impulse of the force, so it can be seen that

impulse = change in momentum

Impulse is a vector quantity with the same direction as the force. It is measured in newton seconds (N s).

The impulse of a force is defined as: impulse = F Δt

Note that N s are the same as kg m s–1 as expected since impulse = Δ(mv), but N s are usually used for impulse.


Force–time graphs


Impulse and collisions

The size of force controls how much damage there is.

Look at the equation for impulse. What determines how much damage is done in a collision, and how can it be reduced?

F Δt = Δ(mv)

Momentum is conserved in a collision, so impulse is constant.

To reduce the force, the time over which the collision takes place should be increased.


Applications

In a car crash, force and therefore damage is reduced by increasing the amount of time over which a collision occurs.

Measures such as seat belts can help with this. The seat belt stretches slightly to prolong the passenger’s impact with it.

Crumple zones also increase the time over which the collision takes place, due to the time it takes them to crumple.


How much do you know about impulse?


Impulse and force–time graphs


Glossary


What’s the keyword?


Multiple-choice quiz

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