https://youtu.be/FZ_-LJjFJR8
PHYS 1101: Lecture Nineteen, Part One
Welcome to Lecture 19. We’re finishing up this chapter in this lecture. I’m going to go over more on these abrupt events and how do you treat them, how do you handle them. By abrupt events, I mean things that happen over a very short time interval. Somebody throws a ball, somebody catches a ball, during the throw, during the catch, a collision with something, something exploding, firing a gun. All of those are abrupt events.
There’s really two perspectives that you can have when dealing with an abrupt event. One is asking: do you care about the forces on one of the objects during this event? Say, on the bullet as it’s fired from a gun, or on the ball as it’s thrown. Do you care about the time duration of the collision? If that’s the case, then it’s this first mathematical tool that you want to use.
It specifies that the combination of the average force on an object during the event times the time, times the duration of the event, that that product sets what happens to that object. It defines a certain amount of change in the momentum that will occur for that object. The momentum, again, is the mass of the object times its velocity. It’s the change. It’s the final momentum minus the initial.
On the other hand, if you don’t care about the details during the event, you don’t care about the force or the duration of the collision, you really want to compare just the momentums that you had before this abrupt event to the momentums that you have after this abrupt event. Then, it’s momentum conservation that’s the useful tool for you.
When can you apply this? This applies to any scenario where during this abrupt event, the dominant force that’s on either of these objects that are involved in this abrupt event is only interactions between them.
When a gun fires a bullet, if the dominant force is the action/reaction pair of the force that the gun puts on the bullet and the bullet puts on the gun, then you can apply this analysis. If, when something’s in the air, and it explodes, if the dominant force again is the action/reaction pair between these two objects, you can apply this analysis.
What it says is that the momentum of all of the objects, usually two, just before the abrupt event, has to stay the same. The sum has to stay the same. Meaning, whatever number you get on the left side of the equation has to equal what the momentum adds up to on the right side, which is just after this abrupt event.
So, these equations that I write here are the simplified versions that we would use for the simple scenarios that we’re going to consider in this class. This first equation represents just one dimensional collisions or one dimensional analysis.
Remember, both of these equations, these are vectors. So, the signs then of these quantities are going to represent the direction in this one dimension. Be it, perhaps, horizontal or vertical. The second equation is also customized, or it’s specific, for only two objects being involved in this abrupt event. That’s also what we’re going to restrict ourselves to in this class.
The other thing we’re going to do today is talk about a special case of collisions that are called elastic collisions. Any collision is an abrupt event, and these two analyses can be applied to a collision.
Momentum is always conserved approximately in a collision, for example, tool number 2 here. But, some collisions are elastic. That ends up giving you additional information that you can use to solve the problem. You’ll see that later in the lecture.
Here’s our first quiz question. Again, it goes over material previously learned in the class. I’ll let you pause the video here and read that. Got another quiz question for you. This is pertinent to the last lecture, where you’re considering impulse and the consequence of momentum change. Before we get into doing a new example, let me say more specific things about the tools that we’re working with, a little bit more on this overview, the big picture.
For this abrupt event, if what you care about is, say, the average force or the duration of this collision for a single object involved in this event, as an example, a tennis ball being hit by a racket. This is the equation that defines the constraints of that, that defines what the logical implications have to be. The left side says that the force times the duration of the collision, that equals, or it causes, it’s responsible for the change in the momentum.
So, here I’ve got some language here to remind you what we refer to each of these terms as. Remember that the combination of the final momentum minus the initial momentum is called the impulse. The impulse is a result of a combination, not just of the size of the force on the object during the collision, but it’s very important that the time be considered. It’s the product of that average force and the time that that force is applied. That combination is responsible for this change in momentum or this impulse.
The other tool is called momentum conservation. The scope of applying this tool is always going to be just before and just after an abrupt event. That could be, for example, the actual process of throwing a ball, of catching it, of collision, of an explosion, of two people on ice skates pushing off from each other.
We did an example of an astronaut out in space throwing a toolbox. Momentum conservation is the perfect tool to use when you don’t care about the details of the forces between these objects during this event. I don’t care what the action/reaction pair was between the two, as this throw was taking place. I just want to know what the final velocity is of one of these objects. In this case, it was the astronaut. Perfect example of momentum conservation being the right tool to use. What are you doing with that? You are setting the sum of all of the momentum before, equal to the sum of the momentum after the abrupt event.
When I break this down further for an event that just involves two objects, which will be the case, I want to say all the time, definitely 99% of the time, the total momentum is only going to be the sum of the momentum for two objects.
When you break that down further, you’ll want to substitute in, for example, the momentum for object one just before the event is going to be m1v10. I have to add to that then, the second object’s momentum. m2v20. That sum has to equal the total momentum after the event. The subscripts do get tedious. You got to take time and be sure you’re not confusing symbols around. I don’t know of a better way to do it to keep track of it all.
I wanted to make a quick side comment about why this momentum conservation is valid. It, perhaps, initially looks like a complicated equation, but once you understand what has to be plugged in for each of these quantities, you understand the concept behind this. It’s really quick to use, and it’s very powerful. You get to bypass having to calculate f equals ma, all these ugly details during this abrupt event. You get to look just at the consequences before, and then what results after.
Here’s my comment, though. Why does this work? The key is you need to think of a group that you’re applying this to. 99% of the time, it’s going to be two objects. What are the two objects that are involved in this interaction, that are on either side of this contact that’s going on during this abrupt event?
If you think about it, at that contact point, the force that each experiences because of the other object, these are an action/reaction pair. These are Newton’s Third Law pair. Newton’s Third Law tells us that these have to be the same size, opposite direction, the whole duration of that collision.
When you write out the definition of the impact on each single object, you’ll see that the impulse that the astronaut experiences has to be equal and opposite to the impulse that the toolbox experiences, because these experience the same force and opposite direction.
In other words, on the left, this is focusing on the astronaut as the object. The definition of impulse is the change in momentum for the astronaut. I know that that impulse is equal to the size of the average force times the time. Well, I’d better make that force negative because it’s to the left.
For the toolbox, the impulse again equals the change in momentum. That impulse is the force on the toolbox times the same delta t, the same duration of this abrupt event. These two forces have to be equal and opposite. Therefore, the impulses have to be equal and opposite. That’s why the overall momentum can’t change.
If one object picks up a little change in momentum because of an impulse, then the other one lost the same amount. So, whatever the number adds up to at the start, the individual values could be different after the event, but the sum has to be conserved, has to be the same.