Mechanics part 3: Momentum
Hey! Welcome back! Last time we talked about Newton’s laws and forces. That’s dynamics, and as we saw, it becomes really easy to find the acceleration of a body using a free body diagram and applying Newton’s laws. But there is one thing I didn’t talk about, and its no trivial matter either! Its something you may have heard of- momentum.
So before we get into momentum, let’s have a quick look at Newton’s 3rd law. This says:
Now applying the second law:
Now this is a very important conclusion. Why? Look at the quantity being differentiated. The fact that the derivative of this quantity is zero tells us that it doesn’t change over time, ie
This is the momentum of the system, and is something we call a conserved quantity, and they’re quite important in physics. It is denoted by the letter p, and as you can tell, it is a vector quantity. In general, the total momentum of a system is simply the sum of the momenta of individual particles. In fact, the conservation of a lot of quantities leads to the discovery of important physical laws. As we shall see later, the conservation of energy directly leads to Newton’s laws. Quite cool, eh?
So what’s the importance of this law we just found out, that momentum is conserved? Well, we can use it to analyze collisions, for example.
In this collision, as you can see, the initial momentum is
As you can see, when the balls collide, they exert forces on each other, which are equal and opposite. This directly leads to conservation of momentum, so we get
So now we have seen how this applies in 1D, but can we generalize it? Let’s consider a system of N particles. Let’s consider all the forces acting on the i-th particle.
So this is the net force on the i-th particle. The internal forces are basically forces which particles within the system exert on one another.
Now let’s exploit something useful. We know that the force particle i exerts on j is the opposite of the force j exerts on i. So when we add these two forces, we should get zero. Basically the sum of all internal forces within a system is zero. We’re gonna exclude external forces for now, and assume the system is isolated. Thus
So we have now not only proved, but also generalized the conservation of momentum. If a system is not acted upon by any external forces, then the total momentum of the system is conserved. But what if there is an external force? What then? Well, we can start by finding the sum of the all net forces acting on all particles.
Now this is a beautiful yet kinda expected result. The rate at which the overall momentum of a system changes over time is just the net external force on it. In the most compact form,
Which, by the way, is the true form of Newton’s 2nd law. It holds even in the case of varying mass (like rockets, for example), and also gives us the impulse-momentum theorem. Basically,
The quantity J is known as the impulse on a body. It is simply the change in the momentum of a body over time. In fact, if you look closely, you’ll find that this theorem is just a fancier way of saying that velocity is the integral of acceleration!
So that’s it for today! This was a long one, but at least now you what momentum is, and why it is conserved! Bye!
Note: I am a high school student and this article is based on my understanding of physics. Feel free to correct me if I am wrong 🙂