Lecture 11: Momentum, Impulse, Conservation of Momentum

Lecture 11: Momentum, Impulse, Conservation of Momentum

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Today, start our discussion of momentum which is, as we will see, a quantity that differentiates objects with the same velocity. Indeed, as we shall see, the momentum of a freight train moving at a speed of 1 mph is quite different from the momentum of a mosquito buzzing around at a speed of 1 mph!
Understanding the concept of momentum will also allow us to study collisions, such as, for example, when pool, or billiards balls, collide. And we will also, later, explore the principles of rocket propulsion.
Today, our focus will be on the definition of the concepts of momentum and impulse, and then we will study the law of conservation of momentum.

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So what exactly is the quantity that we call momentum?
Well, when come basketball season, and our Wildcats are (well, hopefully) playing well, if they’ve won many, many games in a row, you may hear the sports announcers saying UK has a lot of momentum going into the NCAA tournament. They’re hard to beat, hard to stop or change their forward-going trajectory.
Or a little more quantitatively, we could say that a freight train, even if only moving at 1 mph, has a lot of momentum, and is very difficult to stop. Certainly can’t stop a freight train moving at 1 mph with your pinky finger.
On the other hand, the word summertime may conjure up images of annoying, buzzing mosquitoes, and unlike a freight train, it is quite easy to stop a mosquito flying through the air at 1 mph.
And so had a freight train and mosquito both moving at 1 mph, we obviously know which one would be more difficult to stop: the train, and the reason is, because has more momentum!

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Now that we’ve given precise definition for the momentum, and its relation to the kinetic energy, let’s work a conceptual question …

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An important concept regarding momentum is its relation to a net force. Intuitively, we can think of the relation between a force and momentum, by the following logic …
Now we can quantity this with an equation. Let’s start with Newton’s Second Law of Motion, which tells us that the net force is equal to …
So Newton’s Second Law of Motion tells us that if the net force is zero, the object’s momentum does not change. And our “mathematical proof” is completely consistent with our intuitive result, in the box above.

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Now that we’ve seen the relationship between a net force, and the change in an object’s momentum, ready to define the concept of an impulse …

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So let’s work an example combining the concepts of momentum, impulse, and force.

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Now that we’ve introduced the concepts of force, impulse, and momentum, let’s answer, from the physics perspective of what we’ve learned so far in PHY 211, why seat belts and air bags have the capability to save lives, in the unfortunate event you are involved in a serious car crash.
So in the mid-80s, there were a lot of TV commercials aimed at getting people to wear their seat belts. When I was a young kid (well, you know, in the early-80s), people didn’t always really wear seat belts. Wasn’t as much of an automatic thing as it is today. So here’s an example I found on YouTube of such a TV commercial from 1988 (and, yeah yeah, I know some/most of you weren’t even born in 1988…).
So let’s watch this again. What does the seat belt do. Well we see here that the seat belt increases the time over which you come to a complete stop, extends forward, as opposed to you flying forward, and smashing into the windshield, and coming to a complete stop very quickly.
So the point of a seat belt, or an airbag, is that …

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Next thing we want to do is analyze what happens to the total momentum of a system when the objects in the system undergo collisions, such as pool/billiards balls colliding. Here, when we use the word “system”, referring to …

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So to understand what’s going on, let’s consider the collision between two balls, in one dimension. So have these green and red balls, and suppose “before” the collision, about to collide head-on, like so, with these velocities (so speed of green ball much greater than that of the red ball). After the collision, both exit in same direction. If you think about it, this is a plausible scenario …
Now let’s think about what happens when they actually collide. When they impact each other, and are in contact, they exert equal but opposite …

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