Momentum
Introduction to Momentum
· Momentum is a concept that is fundamental to the study of Physics, especially the study of motion, which happens to be very closely related to some of the concepts we have studied already in class (Isaac Newton referred to Momentum as “the quantity of motion.”). It is one of the very basic concepts that govern our universe on both the small scale and the large.
Momentum is a term that is very commonly used in the vernacular; one of the most common usages deals with the arena of sport. It is very common to hear announcers extol the importance and value of a team’s or a given player’s momentum. Generally what these announcers are inferring is that the team or player in question has been doing well and is “on a roll” or is particularly hard to stop. We should all have some inkling as to what this Momentum means. (Indeed, it would not be a stretch to think of Momentum as a different way of looking at the Law of Inertia.)
It will be useful to think of Momentum as being due to an object’s motion (much like KE is the energy we associate with an object’s motion). If an object is stationary, it has no momentum; if an object is moving, it will have a certain amount of momentum. How much, exactly? Well, I’m glad you asked!
An object’s Momentum is simply the product of its mass with its velocity:
p = mv
where p represents Momentum (I know it’s silly and doesn’t make any sense, but that’s the way it is, so let’s just get over it and deal. Gosh!)
Momentum is measured in units of kg*m/s. This doesn’t have a cool name (like the Newton, which is a grouping of other units renamed so it’s easier to deal with; same applies for the Joule).
· One of the things that we need to keep in mind about Momentum is the fact that it is a vector quantity. That essentially tells us two main things:
- There is a directional component to an object’s Momentum; and
- Momentum can be either positive or negative, based on our sign conventions.
· Now I’m sure you’re saying to yourself, “Self, this Momentum thing, p = mv, is a very simple idea. What is Piece all about when he keeps saying it’s so important?” And that’s a fair question; yes, this is a simple idea (or at least it appears so; remember Newton’s 2nd Law, anyone?) that is not mathematically complicated. But it’s a running theme throughout all of Physics that often the “biggest” and most important ideas appear to be deceptively simple and are mathematically quite elegant.
· So how does Momentum get to be complex? Well, the simple fact that it is a vector can introduce certain complications, such as vector components, angles, magnitudes and so forth. (Although, at this point, none of that should be a great challenge to you.) No, where Momentum becomes difficult is when we do one (or both) of the following things.
- Investigate what happens when an object’s Momentum changes.
- Investigate the Momenta of two (or more) objects simultaneously.
A Change in Momentum
· An object’s Momentum will change anytime its velocity changes; notice that doesn’t say speed? That’s because of the vector nature of Momentum. Even a change in the direction of motion will cause a change in Momentum.
We can compute this change rather easily, provided that we have knowledge of the object’s mass, initial velocity and final velocity. With these three pieces of info, we can quickly determine the change in Momentum:
Dp = pf – p0 = mvf – mv0 = mDv
· But hold on a second… Momentum will change when velocity changes, fine, I can see that. But, in order for velocity to change, there must be… an acceleration! (Recall: a = Dv/Dt) And, in order for any object to accelerate, it must experience… a net force!! (Recall: SF = ma) So, what this means, is that there is a way that we can relate the change in an object’s Momentum to the net force that it experiences!
Alright, so let’s think about this. When an object feels a net force, it accelerates. That means, over a period of time, its velocity will change; if its velocity changes, then its Momentum will change. So, basically…, well, here, just check out the math:
Dp = mDv
but
a = Dv/Dt
so
Dv = aDt
and then
Dp = maDt
but
ma = SF
so then
Dp = SFDt
or
SF = Dp/Dt
So this gives us, essentially, the way that Isaac Newton really stated his second law: An object experiences a net force anytime it experiences a change in its Momentum over a given period of time. Or, in order for an object to experience a change to its momentum, a net force must act on for a certain period of time.
The change in an object’s Momentum can be referred to as a quantity called Impulse, J. (Yes, I know this doesn’t make sense either; let’s just go with it and not get caught up on symbolism, shall we?)
J = Dp = SFDt
Impulse has the same units as Momentum. Like Momentum, it is also a vector quantity.
Two (or more) Objects
· When dealing with multiple objects, we are generally concerned with the Momentum of the system, and what happens to it as a result of some sort of interaction between the objects. To deal with this situation, we are going to take a page from our study of energy, and use the concept of conservation again. In other words, when we observe an interaction between two or more objects, we will say that the Momentum for the system is conserved. In other words,
Dp = 0
or
p0 = pf
So, this says that we cannot gain or lose Momentum; it can merely be transferred from one object to another. Another way of approaching this conservation idea is to state that the total Momentum of the system before the interaction occurs must equal the total Momentum after that interaction takes place.
· This interaction is generally referred to as a Collision, even if the objects technically don’t collide. (Imagine a single object that separates into multiple parts; this is technically referred to as a Collision.) It is always assumed, and often not stated, that Momentum is conserved during a collision. Collisions of many varieties will be investigated, but they will all fit into one of three main categories:
o Perfectly Elastic:
§ In this type of a collision, Kinetic Energy is also conserved; i.e., there is an equal amount of KE before the collision as after.
§ These collisions are very rare; the only true collision of this type is one in which objects never actually touch, such as the interaction between charged subatomic particles (where repulsive electrostatic forces prevent them from every coming into contact). However, in practice, the collision between billiard balls is close enough to Perfectly Elastic that we will assume it is so. Also, when dealing with the PASCO© Dynamics carts, anytime the magnetic sides are adjacent, we will assume such a “collision” is Perfectly Elastic.
o Inelastic:
§ This is the most common type of a collision. Only Momentum is conserved. Nothing special here.
o Completely Inelastic
§ This differs from the former in one key area. After the collision, if the two objects become stuck together and henceforth move as one single object (i.e., both objects have the same velocity after the collision takes place) then the collision is Completely Inelastic. An example here is when a football player tackles a man, and the two move after the collision as one new, more massive object.
· Any collision problem will begin with Conservation of Momentum. Based on the type of collision, any special scenarios will be explored (such as the case of Perfectly Elastic, or Completely Inelastic).