KS4 Momentum

Before you tackle this page you must work through the pages on Forces and their interactions and Forces and motion, which are two of the sections of the Forces chapter from the AQA KS4 Physics Specification.

You might also have completed the sections on Work done and Energy transfer, Forces and elasticity, Moments, levers and gears and Pressure and pressure differences in fluids; if not you can do so later.

But now, let's continue with Momentum.

Momentum is a very important topic in the world of Physics; so important that it has its own "Laws", just like those for Gravity or Motion that our good friend Isaac Newton wrote for us.

Well, here's the strange thing. Most books and dictionaries and websites will avoid explaining what "momentum" actually is. That's because it is tricky to put it into words perfectly. Instead they will move immediately to a mathematical definition, which is easy to do and can be done perfectly!

I am going to try to explain the meaning in words and then we will look at the mathematical description.

"Momentum is the property of a moving object that quantifies how difficult it is to be stopped".

OK, so that is just my attempt to put it into words, because nobody else will!

Let's now try to understand what it means.

Consider the following:

I throw 2 tennis balls towards you, one at a time, and you have to catch them ( "stop them").

I throw one fast and the other slow.

Which is the harder to stop?

Pretty obvious; the faster one is harder to stop.

Now I throw another 2 balls towards you; this time at the same velocity, but one is a bowling ball whilst the other is still a tennis ball.

Which is the harder to stop?

Again, pretty obvious; the bowling ball, the one with the greater mass is harder to stop.

What these two simple "thought experiments" tell us is that:

the Momentum of an object depends on two quantities, its velocity and its mass.

- The greater its velocity, the greater its momentum.
- The greater its mass, the greater its momentum.
- If its not moving then it has no momentum.

Well, that's all we need to say about momentum in words; let's now turn our words into a simple equation. I hope you will agree with this:

The beauty of the equation is that we can "mix n match" our mass and velocity values to get a certain momentum answer; we can have slow, heavy objects; we can have fast, light objects etc

So, let's have a go at a few examples.

Before we let you have a go at a few calculations - do you notice how we keep insisting on using the word "velocity" and never use the word "speed" when discussing or calculating momentum?

That is because momentum, like velocity, is a VECTOR quantity.

It has direction as well as size and it is the velocity that determines its direction.

The two tennis balls below are thrown with the same speeds but in opposite directions so they do NOT have the same momentums.

Their momentums are the same size but have opposite directions.

Here is a quick example:

Now its your turn. Use the momentum equation to answer the following questions:

Up to know in our discussion on Momentum we have only been concerned with one object at a time. But something interesting happens regarding momentum when two (or more, but we'll stick to just two) objects interact.

It turns out that when two objects interact, the total momentum that they have before the interaction is equal to the total momentum that they have after the interaction.

Another way of saying this is: In a closed system, the total momentum before an event is equal to the total momentum after the event.

For example, let's say two snooker balls, of exactly the same mass, are on a snooker table and before they interact (the event), ball 1 is moving whilst ball 2 is stationary.

The moving ball rolls into the stationary ball and we have the collision, the interaction or the event.

After the interaction the ball that was stationary, ball 2, is now moving, whilst the ball that was moving, ball 1, is now stationary.

This is a common snooker technique. What you would find is - ball 2 moves off with exactly the same velocity as the original velocity of ball 1; it seems that, in the interaction (the event) ball 2 has taken on the momentum of ball 1.

The original "total momentum" of the two snooker balls has not been lost in the collision but has been passed on or conserved.

We find that this is true for all collisions, interactions, events, even explosions. And it is found to be so universally true that Physicists have declared a "LAW" concerning it; the Law of Conservation of Momentum:

The total momentum before an event is equal to the total momentum after the event.

Or,

Total Momentum Before Event = Total Momentum After Event

Time for a few examples:

We mentioned "explosions" as examples of events. One "explosion" that we will consider is the firing of a bullet from a gun or a rifle.

OK, we have now done an example for each of the common "types" of conservation of momentum questions that you are likely to encounter. Now its time for you to have a go at a few.

When Isaac Newton wrote his Second Law of Motion, which we looked at earlier, his actual statement referred to momentum and specifically to "change in momentum".

What he wrote was more like:

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

Another way of writing this is:

Resultant force = Change of Momentum / Time taken

Or, using symbols:

So, what does it mean?

Well, it simply means that -

when a resultant force acts on a moving object, or on an object that is able to move, then the object's momentum will change and in the direction of the resultant force.

We would see this as a change in its velocity.

For example, when a cricket bat strikes a ball, the bat produces a large resultant force on the ball, changing the momentum of the ball which we see as a sharp increase in its velocity. We also see that the direction of the change in the momentum is the same as the direction of the resultant force.

At the other end of the cricket ground, the fielder raises his hands to catch the ball and he exerts a sufficient resultant force on the ball, against its direction of motion, causing it to have a negative change in moementum, and we see it slow down.

OK, that's a simple statement of what Newton meant. But if we examine the equation in more detail and pay attention to the Δt part, the "time taken", then we notice something a lot more interesting.

Our cricket batsman wants to Change the momentum of the ball as much as possible, so he hits it with as much Force as possible, that is obvious. But look at the equation if we rearrange it for "Change in momentum", mΔv.

mΔv = F x Δt

So, you can see that the Change in momentum is not only proportional to F, which, as we have said, is obvious, but it is also proportional to Δt.

So if the batsman can keep the bat in contact with the ball for as long as possible (a large Δt) then the Change in Momentum will be even greater.

This is why you will ALWAYS see batsmen "following through" with their bat after they first make contact with the ball; they are keeping the bat in contact with the ball for as long as possible and helping to produce the largest possible Change in Momentum. This is why a less powerful batsperson could hit a ball further with good technique compared to a powerful batsperson with poor technique.

The practice of "following through" is also used in golf, tennis (especially on their serves) etc. Footballers also "follow through" when they kick a ball.

So, Δt, the time that the Resultant Force acts, is often called "Contact Time" because in most real life situations that is what it actually is.

To Summarise:

A large Change in Momentum arises from either or both of -

A large Resultant Force,

A long Contact Time.

Now, this alternative version of Newton's 2nd law equation has another surprise for us.

Remember our "fielder" standing far out at the edge of the ground waiting for a ball to catch. If you have ever watched a cricket (or a baseball, or a rounders) fielder, you will notice that when they catch the ball they start with their arms out but as they make "contact" with the ball, but then they draw their arms inwards, taking time to bring the ball to a stop.

What they are doing is increasing the contact time that it takes for their hands to change the momentum of the ball from a very high value to "zero", to bring it to a stop!

Why are they doing this? How does the "long contact time" routine help?

The fielder is the one who is going to apply and "feel" the Resultant Force as the ball is brought to a halt. ( NOTE: A large force will break bones!)

We can examine our equation, back in its first form:

F = mΔv / Δt

Can you see, that to make F, the force felt, as small as possible, the fielder needs to make Δt as large as possible.

So, that is what the fielder does; he or she takes time to catch the ball, sometimes using the whole body, bending the knees to extend the time taken to bring the ball to rest.

The things is, It Works! Newton's Equation Works.

Is this Equation only relevant to sport?

No, of course not. Let's see how it is used in the world of "Safety".

Safety Helmets

You might think that the hard, outer shell of a safety helmet is what makes it really safe, but although it plays a part, it is the thick layer of polystyrene, below the hard shell that actually is the real life saving part of the helmet.

To understand how it does this, you must remember Newton' equation

F = mΔv / Δt

So, in our context, F will be "Force Felt" and the crucial thing to remember is that according to the equation, the "force felt" is inversely proportional to the "contact time", Δt. In other words, if Δt can be made large, then the force felt can be reduced, which is what we would want from a safety point of view.

OK, now let's conside our safety helmet:

In an impact, this thick polystyrene layer takes time to become squashed, so Δt is large.

And if Δt is large, then the force felt is reduced. Thats it.

So, all safety helmets work by having a relatively soft squashable layer of polystyrene which will deform slowly on impact and because it deforms slowly, the force felt by the wearer is reduced.

(NB 1: A thick SOLID helmet would not be good for surviving crash impacts because there would be no squashing and so the contact time would be minimal and therefore the force felt would be a maximum.

NB 2: If you compare a modern soldier's helmet to that of a WW2 soldier you will notice that the modern helmet is much bigger because it not only has the steel outer for bullet protection but it also contains a softer interior for impact protection; the WW2 helmet was a simple steel shell.)

Well, thats Safety helmets done, so now we have to discuss safety flooring, then crumple zones on cars, then air bags etc.

But the good news is, all of these "work" in exactly the same way as the safety helmet. We will just mention each one briefly:

- Safety flooring in playgrounds - when a child falls, the floor squashes (like the polystyrene), taking time to compress and so the force felt is reduced.
- Crumple zones at the front and back of cars are deliberately designed to "crumple" or squash readily on impact, taking time to compress and so the force felt (inside the car or outside the car) is reduced.
- Air bags - inflate immediately following a crash; a head moves into one, taking time to come to rest as the bag "squashes" and so the force felt by the head is reduced.
- Seat belt - slightly different from all the above - seat belts work by taking time to stretch (rather than to squash), but still increasing the time it takes to bring the person to a rest and so reducing the force felt.

Now its time to have a go at a few example calculations using the equation that we have been exploring.

Its about time you had another go at some questions by yourself; let's see if we can include some "word" questions amongst the usual calculations since examiners will expect you to be able to explain how all those saftety features work.