9.3 1D Collisions

Today, we're gonna look at applying some momentum to collisions. Today, we'll focus on one-dimensional collisions, and later, we'll look at the angles and vectors involved in two dimensions. The idea is that we can measure the momentum of objects before and after they collide to figure out missing things like speeds and directions. So, we'll get a few examples in a minute, but the key ideas are: when we're talking about collisions, we're talking about two or more objects striking into each other. We'll do it in one dimension today, but things can be like head-on or one object hitting the other from behind. Both objects can be moving or one can be stopped, it just depends on the scenario. The key idea is that when they collide, they're gonna exert a large force over a short period of time on each other, which means if we change the In this lesson, we're going to change the momentum, which will result in an impulse. We will only focus on linear collisions or one-dimensional collisions for today. Back in Nuuk, this time they noticed that the total momentum before the collision of all the objects (there are two objects being better climbing the tube) is equal to the sum of the momentums of each of them added together before the collision. This also holds true for the total momentum afterwards. We basically end up with the conservation of momentum. In Physics 20, we don't discuss conservation of energy. Instead, we will look at conservation of momentum. When objects collide in an isolated system, meaning that no energy enters or leaves, and all the energy stays constant, the total amount of momentum is conserved. This implies that there are no other external forces involved. Acting in a play, if that's the case, our total momentum before and after is conserved. So the totals before and after are the same. Let's look at how some formulas are going to be used for this. Before the collision, we would have object A with momentum (PA) and object B with momentum (PB). The total momentum before and after is going to give us some sort of total. After the collision, the momentums will change, so we'll use primes (PA' and PB') to represent the momentum of object A and object B afterwards. The formula states that the total momentum before has to equal the total momentum afterwards. This means that the sum of PA and PB before the collision should be equal to the sum of PA' and PB' after the collision. Are the same, so the momentum of A and B added together equals that of A and B afterwards. They are the same totals. So we can even break this down further if the question is dealing with masses and velocities. We could say the mass and velocity of object A plus the mass times velocity of object B has to equal the mass of A and velocity (I guess the mass will stay the same, we're typically not changing the masses) for the mass of A afterwards to be the same. And then we have velocity of A afterwards will be different, and then we have mass to be in velocity. Okay, so that's basically how we're going to deal with these kinds of questions. So you're going to use either one or both of the plenty. Usually we use the mass and velocity, and that's the most common because typically the questions are. Gonna be saying things like "What is the final velocity of object A?" or something along those lines, so let's do a couple of examples. The first example is where we're going to have two objects that are going to strike and bounce apart. The second example is going to look for two objects that strike but stick together. Both are similar, but you have to be a little bit careful when you're solving them. So, this first one is where we've got two pool balls. We've got a billiard ball moving to the right. It hits another one that's stopped, and the question is, what's the velocity of the red ball afterwards? So, we set this up. We'll use colors so we can keep track of this a little bit better. We're going to have the mass of A times the velocity of A plus the mass of B. The paragraph should be corrected as follows: The mass and velocity of B before the collision will equal our mass and velocity after the collision. Let's substitute it in. The mass of the first one was 0.15 and its velocity was 8. The mass of the red one is stated as similar, so it'll be the same size, 0.15. But its velocity will be zero ("rest"). That means the whole thing will be zero. On the right-hand side, we have the same mass, but this time it says it's moving to the right. We have to be careful and consider that as positive. So our original velocity was... Positive eight. It's still moving to the right, so that would be positive 2.5. And then our red ball has the same mass, 0.15. The question is, what's its loss? So you can see we plug everything in the formula. We have everything except for the final velocity of ball B. So now all we have to do is do the math and solve for all this. So, if we go 0.15 times 8, it gives us 1.2, and that's our momentum. We don't have to buy units for this one, it is zero. So we're not adding or subtracting anything. And now we go 0.15. Times 2.5. That gives us 0.375 plus 0.15 BB prime. So if we do the math, let's subtract 0.375. So, 1.2 minus 0.375 gives us 0.825. And we're going to divide by 0.15. That gives us a final answer of 5.5. And since we're looking for velocity, that is meters per second. And if we can't be careful with the symbols and the directions of positive and negatives, this works. That means we got a positive 5.5. So, that means It's going five point five to the right, right? Because we called radar positive direction, so it's also gonna be going right as well. Let's do another example now, where the objects stick together. So in this case, we got two different masses, and they're going to collide and stick together. Let's go through the same formula again: A + B = A afterwards + B afterwards. Okay, so the only thing that's a little bit different in this case is when we're solving this, the two masses were separate at the start, but then they combine afterwards, so they're going to combine together and we'll just be one. We're just gonna have one big mass, let's call it "massive A and B together," and it's just gonna be moving together as one speed. So, that's the only thing that changes with these when they stick together. We're just gonna combine the two A and B's into one. You can write it as A+B or just AB. It doesn't really matter. You don't need to worry too much about the symbols. So, in this case, we have to be careful. We've got a blue one that's 2.3 and it says it was moving to the right. So, 2.3 times 12. And it says we've got a red one that is 5.6 and it was moving to the left of 8.1. So, we're gonna put that in as negative. So, be really careful with positive and negatives. Now, afterwards, the masses are gonna stick together. So, our total mass is gonna be the same, but it'll be 5.6 plus 2.3. So, that will give us 7.9. Nine for a total mass, and the question is what's its total velocity afterwards? So, a pretty basic question. All we have to do is figure out the left-hand side and divide by 7.9. I won't bother doing the work on this one. When we plug all that in on our calculator, we should get an answer of -2.2. Okay, so we have to be careful. Let's think about the logic a little bit with this one. We had a smaller mass going to the right at 12, and we had a larger mass going to the left at 8.1. This one's quite a bit bigger, but it's going a little bit slower. In essence, though, the total momentum is still bigger, so V has a bigger momentum going in. So when they collide and stick together because it has a larger momentum, it's going to carry the mass to the lower left slower. So that's what the negative is telling us, that after they collide, the total mass is going to go to the left. So when objects stick together, you can see just a slight modification of the formula: have the masses together and then you just have one mass and one velocity. It's a common question because quite often we'll have cars or trains or something colliding and they get locked together, so they'll move together as one object. It's also possible that the objects could be stuck together at the start and then they separate and go into different pieces afterwards. Typically, a question like that would be some object that explodes. So let's suppose we have a box that has a certain mass and it's at rest and it blows up into three different pieces that go in different. Directions and vectors are closely related. We still perform the same types of calculations, but typically we encounter them in two different scenarios: directions and vectors in two dimensions. In that case, we will save those types of questions for the next lesson. So, that's all for today.