Momentum and Force

What is Momentum?

Momentum is a measure of 'oomph'.

 An object with lots of mass will have more momentum than a lighter one travelling at the same speed. Similarly for two objects with the same mass, the faster one has more momentum. The instantaneous momentum of an object is given by the simple equation:

p = mv

p = momentum

m = mass

v = velocity

[NB: an underlined quantity indicates that it's a vector, i.e. it has both magnitude and direction]

Momentum in what direction?

Momentum is a vector, which means that it has a direction. So if you're talking about an object's momentum, it could well be important to specify which direction that momentum is in. For example, this object has some velocity v at some angle θ to the horizontal.

You could say that this object has momentum mv at the same angle to the horizontal. Alternatively you could split it up into component parts like this:

Now we would say that the object has mvsinθ up-down momentum and mvcosθ left-right momentum. This is called expressing the vector in component form, because there is an up-down component and a left-right component.

It is perfectly possible to have negative values for momentum; all it means is that your object is travelling backwards. So if I defined 'North' as positive and an object has -x momentum, that means that it has x momentum South.

Momentum Changes

You will have heard that momentum is always conserved. I'll show you the very simple derivation of that law in a moment. For now I want to go through Newton's 1st three laws of motion and clarify what they all mean.

Newton's 1st

"If there is no net force on an object, then its velocity is constant"

This one is fairly intuitive. If you don't push an object in any way then it will not change the way it moves. It is worth noting that this is where Newton deviated from the tradition. It had been accepted for thousands of years, since Aristotle, that a force is necessary to keep an object moving and that objects will tend towards being at rest if left to their own devices. The reason they thought this is that they could not see lots of hidden forces which do tend to slow objects down like air resistance and friction. If you took those away you would see the truth of Newton's 1st: no force means no change in velocity.

Newton's 2nd

"Force is proportional to the rate of change of momentum"

Velocity is just how much your position in space (your displacement) changes in a given interval of time. Similarly, force is how much your momentum changes in a given interval of time. The faster your momentum changes, the more force you will feel. The less force you experience, the less your momentum will change per unit time. So force can be expressed as follows:

F = dp/dt = d(mv)/dt

F = force

p = momentum

m = mass

t = time

If mass is constant, which it usually is, then it can be taken outside of the derivative and we get:

F = m dv/dt 

F = ma

a = acceleration

So the more massive an object is, the less it will accelerate under a given force.

Once again, force is a vector, which means we have to specify a direction. Remember our particle from earlier? If we apply a force in the -ve left-right direction then his momentum will change. But because our force is only in the left-right direction, its up-down momentum will remain unchanged, no matter how much force we use or for how long.

If this force is applied for some length of time then the velocity of the particle, expressed in component form looks like this:

You can see that our left-right velocity has changed to A, but our up-down velocity has not. So how do we calculate A? Remember back to Newton's 2nd law: F= dp/dt. If I told you that a car was travelling at 10 m/s for 30s then how would you calculate it's displacement? Simple, you just multiply the car's velocity (it's rate of change of displacement) by the amount of time it was travelling for. Similarly, to calculate how much was the momentum change of our particle just multiply the force (it's rate of change of momentum) by how much time the force acted for:

ΔP = Ft

So our change in momentum is equal to Ft. A specific change in momentum is also called an impulse. Now, back to calculating A. A is our final velocity, so our final momentum must be equal to Am. We also know that this quantity Am is just our initial momentum combined with the change in momentum. Since our change was negative (in the opposite direction to where we were initially traveling)  we can calculate Am by subtracting the change in momentum from the initial momentum:

Am = mvcosθ - Ft

Plug in values and you can calculate A! Having done this you can go back from component form to a single resultant vector.

It is also possible to solve this kind of problem geometrically using something called a momentum triangle. Write down an arrow for your initial momentum, tack on another arrow for your change in momentum (or impulse) and wherever you end up will be your resultant final momentum.

Newton's 3rd

"To every action there is an equal and opposite reaction"

Simply put: if you push me, I push you back. For any force (and that is ANY force) there is a reaction force to it acting on the other object in the other direction with exactly the same magnitude. Let's look at an example:

I'm currently sitting on a chair, so the chair is pushing up on me. What's the 3rd law pair? It's the downward force that the chair feels from me. Similarly the chair is standing on the ground with the ground pushing up on it. What's the 3rd law pair? The downward force which the Earth feels from the chair. 

A question you may well be asking is 'If every chair on the earth is pushing it down then why doesn't the Earth accelerate off in some direction?'. The answer is still Newton's 3rd law. Consider my weight. It's a force which acts downwards on me. So what's the 3rd law pair to this force? In this case it's the gravitational pull of me on the Earth upwards. Take note: the force acts on the Earth, not on me. 

If the chair and I stay still then the forces on us must be perfectly balanced (remember Newton's 1st). This means that the force on me upwards from the earth must be exactly equal in size to the force of my weight on me downwards. But similarly, the 3rd law pair forces on the Earth will be balanced: the downward force from me and the chair will exactly equal the upward pull of gravitation on the Earth. So the Earth doesn't accelerate off in any direction because those pushing forces from all those people sitting on chairs will be exactly equaled by gravitational pulling forces from those same people.

If the forces on me aren't balanced, say I've jumped and accelerated up, then the Earth will accelerate a bit. But remember back to Newton's 2nd. F=ma. The Earth is incredibly massive, so that little bit of extra momentum that I give it will not accelerate it by very much.

Conservation of Momentum

Momentum is always conserved. Whenever the momentum of one object changes, the momentum of some other object must change to accommodate this. If it ever looks like momentum isn't conserved then it's because you're looking at a non-closed system.

I'm sitting in my car and I start moving. I've gained some momentum, so something else must have lost some momentum. But it looks like nothing else has changed in momentum, so how has momentum been conserved? The answer is that the Earth will have sped up in the opposite direction. If we just look at the car then momentum is not conserved because it is not a closed system as there are forces external to the system acting on it. If we consider the Earth and the car then momentum is conserved as the two taken together form a closed system.

But why is momentum conserved? It comes down to Newton's laws. The 2nd law shows us that, in a given stretch of time, a force is equivalent to a change in momentum. The 3rd law shows that a force in one direction will be exactly balanced by another force in the opposite direction. But if a force is equivalent to a change of momentum, this means that any change of momentum in one direction will be exactly balanced by a change of momentum in the opposite direction. This means that the overall momentum of a closed system never changes: momentum is conserved. Done mathematically:

If body A exerts a force F on body B for t seconds then body B will gain Ft momentum.

By Newton's 3rd, body B also exerts an opposite force on body A for the same length of time. So body A will gain -Ft momentum

The total change of momentum for both bodies is given by the sum of their individual changes:

Δp = Ft - Ft = 0
So the overall change in momentum is always zero for any interaction between bodies.