3.4.1.6 Momentum
$momentum=mass\times velocity$
Conservation of linear momentum.
Principle applied quantitatively to problems in one dimension.
Force as the rate of change of momentum,
Impulse = change in momentum
$F\Delta t=\Delta mv$, where F is constant.
Significance of the area under a force–time graph.
Quantitative questions may be set on forces that vary with time. Impact forces are related to contact times (eg kicking a football, crumple zones, packaging).
Elastic and inelastic collisions; explosions.
Appreciation of momentum conservation issues in the context of ethical transport design.
Momentum and its conservation
Momentum is an idea that should be familiar to you, and the basics of it are very simple. Momentum is the product of an object’s mass and its velocity:
Momentum can be imagined as the un-stopability of an object. This can be visualised by thinking of a small child running and a large rugby player running at the same speed. The child has much less momentum than the rugby player and is therefore much easier to stop.
If this were it then momentum would be quite a boring topic, fortunately momentum is much more important than just measuring un-stopability. Momentum is a conserved property. This means that when any two (or more) objects interact with each other momentum is conserved. Or, the total momentum of the objects before an interaction is equal to the total momentum of the objects after the interaction. This is very important because this applies to ALL interactions, collisions, explosions, etc.. This can be shown mathematically as:
If we imagine a simple situation (and one that you will see in class), where two trolleys with the same mass collide and stick together, one being stationary and the other moving, you can intuitively guess that their final velocity will be half the initial velocity. And it is easy to see that as the mass of the moving system doubles, then the velocity halves. However, the situation below is a little more complex. This law applies to closed systems with no external forces, such as friction, acting.
Here we have two trolleys with different masses, moving at different velocities. The quicker trolley will catch up and collide with the slower one and stick to it. So what speed do they move off at? Do they slow down or speed up? Applying the conservation of momentum:
The sum of the two trolley’s momenta before the collision is:
So this is also the value of the momentum of the two trolleys when they are moving together, so rearranging the equation for momentum we can find the velocity of the two truck moving together:
This is fine for moving objects that already have momentum, but what about when objects are stationary and have zero momentum? In the diagram below the two trolleys are not moving but are about to be pushed apart by a spring placed between them. If they had the same mass then the spring would apply a force to each of them which, according to $F=ma$ would produce the same acceleration. But in this case they each have different masses, so we can derive an expression for the relative velocities of the two trollies.
The initial momentum of the system is zero:
The two m terms cancel out to give:
This tells us that the lighter trolley has the greater velocity, and the minus sign tells us that they are both in opposite directions.
This idea was taken tested in this hard exam question from 2016, to see it worked through click on the link below.
Elastic and inelastic collisions
So far we have only looked at simple explosions or ‘sticky’ collisions, but of course objects can bounce off each other as well as stick. Whatever happens to the objects in the collision, momentum is conserved (so long as the friction is negligible). Is energy always conserved in these interactions? Of course the total amount of energy is conserved, but the objects in the interaction are not a closed system, and energy can be lost as heat, sound or internal energy. In these cases the kinetic energy of the system decreases as a result of the interaction. Springy collisions are properly called elastic collisions and these conserve both kinetic energy and momentum, whereas sticky collisions are inelastic, and lose kinetic energy after the collision.
| Momentum | Kinetic Energy | Total Energy | |
|---|---|---|---|
| Elastic Collision | Conserved | Conserved | Conserved |
| Inelastic Collision | Conserved | Not conserved | Conserved |
To understand how momentum can be conserved when kinetic energy is lost we need to look a bit more closely at the equations of momentum.
When two objects interact, their individual momenta change, and that is usually because their velocities change. So the change in momentum can be written as:
We can see the term $(v-u)$ in the equation for change in momentum, which, if divided by $t$ will give acceleration, and in this case is equal to force:
This means that force can be defined as the rate of change of momentum (Newton’s II law) and that. We can also see that $\Delta p=Ft$, so the change in momentum of an object is related to the product of the force applied and the time it is applied for, whereas the energy is related to the force applied and the displacement it causes. Therefore energy and momentum are have different dimensions, those of time and length, therefore it is possible for one to be conserved when the other is not. The product of the force applied and the time is called the impulse of the force.
Looking at the force-time graph below we can see that similar to the force-displacement graph the area under the line is equal to the change in momentum of the object.
This relationship between force, time and momentum is utilised in many safety products such as, car crumple zones, bubble wrap and trainer cushioning. You will research this both theoretically and practically in class.