9.1 Momentum and Impulse

Ok, today we're going to start our first set of notes on Unit 1 on momentum and impulse. We'll go through what those are and what they mean. Today, we'll also look at a few formulas and some examples. To begin with, just to get you thinking, let's ask ourselves: How does a karate guy chop through a cement block with his bare hands? Or why, when you fall on a trampoline, does it hurt less than when you fall on the floor? Or why do people in a big bus or semi not get injured in a car accident compared to somebody in a small car? Those kinds of questions are common physics questions that we try to wonder why, why it works the way it does. And today we're gonna figure out that the reason that happens basically is due to momentum. Momentum is sort of a weird topic. It doesn't really have a clear definition, something like velocity or distance or anything like that. It's sort of an idea that combines mass and velocity. So, if you think of when we did Physics 20, we talked about inertia and kinetic energy. It's similar to those. So, if you have something that's really massive, like a big, huge semi-truck, and it's moving at a fairly fast speed, it's gonna have a lot of momentum. Whereas, we could have a little tiny car going at the same speed, but because it has less mass, it's gonna have less momentum. So obviously, inertia plays a part. If something is at rest, it's going to want to stay there. If it's moving, it's going to want to keep moving. And also, it has a lot to do with kinetic energy. If it has lots of energy, typically, it's going to have lots of momentum. So, we get this velocity and mass combination at play. What we find out, though, is momentum isn't really a, it isn't really either mass or inertia or kinetic energy. So, what we're going to do is kind of compare and contrast the two together. The first thing you can see is, when we compare momentum and kinetic energy, they both have mass, they both have velocity. The key difference, though, is kinetic energy. Remember, energy and work is similar. So, kinetic energy means it has the ability to work. Momentum can't do work. It just has its mass and velocity, and that's it. And when we compare it to inertia, if we have something that has a large or small amount of momentum, that's gonna mean a high or a large or a small amount of inertia, complete, you know, whichever. So, that's the big truck has lots of momentum, it's gonna have lots of inertia. A little tiny object, it's not moving very fast, and it has a little bit of momentum, it's kind of a little bit of inertia. And the biggest difference is inertia is just a definition or a concept. It's not something that we can actually measure with the formula. So, our first formula that we're going to use when we're dealing with momentum is P = mv. And P is the symbol we use for momentum. Not really sure why they chose P. I guess mmm is already used for mass and things like that, so we try to use a letter that's not going to be used. So, we use P and mass, a normal mass in kilograms, velocity or normal speed in meters per second. And with momentum, when we multiply those two, we get kilograms times meters per second. So, when you see here, the key thing to remember is momentum has no special derived unit. Remember back to force, we said things were in Newtons. When we talked about energy, we said things were in joules. Those were all derived units, named after somebody. Momentum doesn't. It's just momentum, and we go kilograms times meters per second. Let's do a couple of examples using that formula. So, our first one, we have a thousand-kilogram car moving at 10 kilometers per hour. So, we just want to use P = mv. So, pretty basic, we have 10,000 kilograms times velocity. But remember, velocity has to be meters per second, so you have to divide 10 by 3.6, and that gives us 2.2. I've been repeating, so the best way to do this on your calculator is to type in a thousand times 10 divided by 3.6, and you don't have to worry about rounding the decimals or anything like that. So we get an answer of 2777.7, and we'll round it to two sigdigs because in our question, remember, we have to get it. So we'll round up to 2.8 times 10 to the 3 kilograms meters per second. Okay, so just like in physics, honey, we always want to not round any decimals until the very end and then round it correctly to the right number of significant digits. Okay, the next example I have here is one that we've sort of looked at before in physics 20 as well. And this one is, we have a meteor. It's got a momentum of 1.13 times 10 to the 4. So e to the 4 remember just means times 10 to the fourth. So what's gonna happen is it's gonna slow down and it's gonna lose its mass. So remember when we deal with these, we're still dealing with the same formula, but now our momentum number, whatever it was, that was 1.13 times 10 to the fourth. If we're gonna go a quarter of the speed, that means we're gonna have to multiply that amount by 1/4. And if we shrink the mass by 2/9, we're gonna have to multiply it by 2/9 as well. So basically, we're just gonna be taking our original momentum and multiply it by 1/4 and then multiply it by 2/9. That's the easiest way to do these questions. So when we do that, we actually get a final answer of 627.7. So we rounded off on the third digit, so it's 628 kilograms meters per second. Okay, so we did these kind of questions in Physics 20. And if you can remember, we did things like double the mass, triple the distance, those kind of things. It's the same sort of idea, just using a different formula. Okay, our next definition for today is impulse. An impulse is basically just comparing what the change of momentum is. So if a car is stopped and it's gonna speed up, it's gonna have a momentum of zero up to whatever its momentum is going to be. Whenever we have its mass times velocity, so momentum is changing, recall back impulse. So just there, a fancy word meaning change of momentum. But a couple of key things to be aware of here. So if impulse, this means momentum is changing. That means the object must be moving at different velocities. All right, so the object's moving at a different velocity. We typically assume the mass stays the same. Right, if we're driving a car, our mass, the car isn't going to change. All we're really doing then is changing our velocity. So what we do then is our formula then gets altered a little bit. So instead of just being p equals mv, we now have momentum changing. So we use the little delta sign for momentum. Mass stays the same. And then our velocity changes to delta v. So remember a couple of key things. So remember change momentum or delta p is going to be our impulse. And then delta v, just like before, that just means our final minus initial velocity. Okay, so let's try this next example. So we have Box of Tic Tacs, weighing 15 grams, slides along the table at 5 m/s. Pam reaches down to slow it down, and it eventually only moves at 1.6 m/s. The question is, what is the impulse? If we use our new formula, where the mass is 15 grams (or 0.15 kilograms), and the change in velocity is from final minus initial (1.6 - 5), we have: Impulse = (1.6 - 5) * 0.15 = -0.51 kg·m/s Since momentum is a change, it can be positive or negative. A positive value means we are gaining momentum, while a negative value means we are losing it. Now, let's link this to Newton. If velocity is changing, it means the object is accelerating. And acceleration can only occur if there's a force acting on it. Therefore, force must be related to impulse. This is where Newton's second law comes in. We used to write it as force equals mass times acceleration. But in reality, Newton's original formulation was that force is proportional to the change in momentum divided by time. He said that the more momentum changes per second, the greater the force. From there, he derived the formula force equals mass times acceleration, by substituting momentum with mass times velocity, and velocity divided by time with acceleration. So, we can use both formulas: momentum equals mass times velocity, and force equals momentum divided by time. Let's rewrite them a bit. Momentum (m) = mass (m) times velocity (v)Impulse (I) = force (F) times time (t) Since they both represent momentum, we can combine them:m * Δv = F * Δt This new formula allows us to solve problems using any of the three variables: mass, velocity, and force. To continue with the question, let's suppose I give you the mass, velocity, and time. And it's asking for the force. We can use this new formula to solve it for F, so you can solve it for any of the four as long as you have the three other points. For example, we might just use the left side to find impulse if we're given mass and velocity. Or we could use the right side to find the impulse if we're given the force and time. So, depending on how the question is worded, you can sort of pick which part of the formula you want to use - all of it or just one of the two sides. They're all dealing with impulse, so that makes it a little bit handy to work with. So let's do this next example. We have a rifle shooting the bullets, so we have a mass, we have time, and we have velocity. The question is asking, what is our force? We're going to use this new combination formula and solve it for X. Our mass is 9 grams, let's put that in kilograms - 0.009. Our velocity started at 0 and it's leaving at 200, so we just have 200 - 0. We can put that in if you want or just leave it as 200, whichever. And then that will be force times time. Time, same thing. Our time took 3 times 10 to the negative 2 seconds, and it started at time 0. So we can put the zeros in if you want or not, and maybe it's easier just to leave the - zeros out. So now to solve it for force, just rearrange the equations: we'd have 0.009 times 200 divided by 3 times 10 to the negative 2, and that gives us an answer of 16, so we rounded off to three significant figures and wrote it as 60.0 Newtons. Okay, just a quick example to show you how you can use this new formula by combining both sides at once. Let's talk a little bit about momentum and force and all those kinds of things, just to make sure you understand what's going on. So with our formulas, let's look at just the first one. Our momentum formula or the impulse formula was force times change in time. That means we could have a certain force and a certain time, and that would give us our answer of momentum. Okay, so let's suppose we had a medium force in a medium amount of time. Now, what would happen if we make the force bigger? So we use a large force, but then we use a smaller time. You can still get the same answer, right? This is sort of like saying if our force was 5 and our time was 2, they'd have an impulse of 10. But now, if we made our force 10 and our time 1, it's still 10. So we can have lots of force and a little bit of time to get the same answer, or we could do the reverse. We could have a small little force and a huge amount of time, and that would also give us the same answer. So let's suppose their force was only one, but now we're going to change their time to 10. You still get the same impulse, alright? That makes sense, doesn't it? Because we're dealing with force, sometimes this goes into a lot of engineering to help us deal with things like collisions. If we can increase one and decrease the other, then sometimes that'll help. Usually, we can't change momentum, right? So mass is going to be the same and our velocity, like if we're driving in a car and we're driving 100 km/h, and then come to a stop, our change in velocity is still going to be 100. Our mass is still going to be whatever it is. So really, the only thing we can sort of do is change the amount of force and the amount of time that act on the car. This is why, if you think in terms of collisions, if you're driving a car, you'd rather hit a haystack as opposed to a brick wall. Alright, we have the same mass and velocity, but by hitting a brick wall, we're going to have a large force and a small time. So that's going to give us lots of force because we have lots of force over a small amount of time, right? So that's going to create the same momentum again, but lots of force. If we hit the haystack, then obviously the time is going to be bigger. So then we're going to have less force and a bigger time. If we get a longer time period, then our impulse gets spread out, meaning our force then has to be smaller. So that's the whole idea behind the airbags in a car, right? If we didn't have an airbag and hammered on the brakes, we're going to have lots of force in a short amount of time when we hit the steering wheel. But if we have the airbag there, that airbag delays the time. So we get a longer time period, which then reduces the force. Remember, our momentum never changes. Our impulse is the same. Momentum and impulse mean the same thing, again. So those aren't going to change. The only thing that we can sort of adjust is our time. If we make our time larger, the force will get smaller, which is a good thing. Okay, so that's why when we use a bungee cord, it's going to stretch out the time. We have less force. Or even if you fall on a wooden floor, or a grass field or something like that, it doesn't hurt as much because the time, because that floor, the grass sort of bends a little bit. It sort of delays the time. It hurts a lot less than if we were to fall directly onto a concrete floor. So let's do one more example, and then that'll be it for today. So here we have a 75-kilogram man. He's in a car accident. He's traveling at 65 kilometers per hour, and he's going to come to a stop when he hits a truck. Part A is to calculate how much force was on his body if he has no airbag and he just comes to stop in half a second. So we're going to use our new formula again. Force times time equals mass times velocity. Force is what we're looking for. Our time is 0.5 seconds. We have a 75-kilogram mass, and our change in velocity is from 65 to 0. Remember, we want to change 65 into meters per second. Let's divide that by 3.6 so we can do all of this in one shot. So we have 75 times 65 divided by 3.6, and we're going to then divide it by 0.5. When we do this, we get a force of 2708 Newtons. And if we round it to two significant figures, it would be 2.7 times 10 to the 3. Pretty big number. Okay, so now let's do the exact same question, but let's suppose the airbag now is just going to delay the time. Instead of going from half a second, it's going to go to 0.78 seconds. So the rest of the question stays the same. Let's do this calculation, and we get an answer of 1736. Or if we round that off, that would be 1.7 times 10 to the 3 Newtons. Okay, so how can we compare these two forces? We can see that there's about a thousand less Newtons. Might not sound like a lot, but maybe that's enough to keep the guy from being killed. We could even kind of do a comparison. If we take 1.7 times 10 to the 3 divided by 2.7, we can see we get about 60%--63%. Percent protein: Okay, so there's quite a bit less force when we have a shorter time. With more time, there is less force. There are a couple of key things to remember. First, technically, our force is considered negative because our speed is going from 65 to zero. So in our formula, we would actually have 0 minus 65, making the 65 negative. This negative value indicates that we have a negative force, which makes sense since we are decelerating. However, the negative sign doesn't make much of a difference. I also noticed that if you compare this to the note page, there is a table. In the notes, I have the time as 0.05 seconds, which actually resulted in a force of 7 x 10^4 without the airbag, which would obviously be a lot bigger. But the calculations are the same. That's it.