This is a toy called a Newton's cradle. Usually you see them about yea big. But it's almost like magic because you lift the ball on the side, and then these three stay still. And this one suddenly flies off.

(Describer) Hanging metal balls on the ends swing one at a time. She stops them.

How does this happen?

(Describer) When she swings *two* on one end, two on the other end swing.

This is so cool. This is also a really big Newton's cradle. No, this is a big Newton's cradle.

[epic music playing]

(Describer) Outside, bowling balls hang in a line.

Hey, I'm Dianna Cowern. Welcome to lesson 13 of Dianna's intro physics class, also known as AP Physics I review, also known as Physics by Dianna. Today's lesson is about things that smack into each other. Put more properly, it's about objects that collide. And we want to look at what happens to the motion of those objects after they collide. So the theme of today's episode is "Star-Crossed Lovers." When Romeo and Juliet's worlds collide, will they stick together and live happily ever after, or will they tragically part ways after their brief encounter?

(Describer) She holds a toy cow and cat.

Spoiler alert, it's not a happy ending. So what you will know by the end of this lesson is that the correct tool to use when you're solving problems about collisions is conservation of momentum. And lucky for you, we just did that in the previous lesson. So that's the big idea for today. Plus we're going to talk about the types of collisions, like the sticky ones, like a pie sticking to your face--

[squishy sound]

--or the bouncy ones, like a tennis ball hitting a racket. And we're also going to figure out which collisions are which type. So let's do a little quiz, sticky or bouncy, or in physics terms, inelastic or elastic? I always have trouble remembering which is which. So if you think of elastic as like a rubber band, and rubber balls bounce. So elastic collisions are the ones where things bounce off each other. And it doesn't always happen with rubbery things-- just putting that out there. Inelastic are the collisions where they stick together. And we'll get into more useful detail about what qualifies them as elastic or inelastic. But first, the fun quiz. Asteroid hitting Earth, inelastic or elastic? That's inelastic because they stick together. The asteroid ends up embedded in the Earth. Like a person jumping into a pool of Jello, a spitball hitting the kid in front of you? Correct, inelastic. What about pool balls smacking into each other? That would be elastic. The balls bounce off each other. And even though they're not rubbery, that's elastic. What about a small water balloon hitting you in the face? That's a trick question because it depends. The balloon could stay intact, in which case it'll bounce off, and that's elastic. Or it could break and soak your face, and that would be inelastic. How about the Newton's cradle? Elastic. Good job. Bumper cars gliding? Elastic. Real cars crashing into each other? Most likely inelastic. And here are some can go guess on your own--subatomic particles colliding, black holes colliding, dropping a ball onto oobleck. I like that one. Anyways, collisions are huge in physics, metaphorically. And the big idea of the day is that, as always, during a collision between objects, conservation of momentum holds. The total momentum of the system before and after the collision is the same. And that's going to be our main tool today when looking at types of collisions. So let's look at inelastic collisions first. Two objects collide, and they lock together and move as one. The classic example that's going to help us wrap our brains around them is trains, like those little toy trains with the magnets. Imagine two train cars rolling on a track and then hitting and linking up. It's classic because it's simple. So let's model it.

(Describer) She pulls out some paper.

Give me a good arc.

(Describer) She catches a marker.

Hee! So, two train cars on a track.

[humming]

(Describer) She draws them.

We'll say that their masses are m1 and m2. One of them is still, so v2 equals zero. And the other one is moving with a velocity v1. So then they collide, and they stick together.

(Describer) The paper rolls up.

Wow!

(Describer) She unrolls it.

So they collide, and they stick together, and they begin moving as one object with a new mass and m1 plus m2 and a final velocity vf for v final. Here's the question that you can solve in a problem like this. Given the initial masses, m1 and m2, and the initial speed, v1, what will the final velocity, vf, be? Like we said, our key idea is conservation of momentum. If we add up the total momentum of the two cars before the collision, that's going to be the total momentum of the coupled cars after the collision. Because one of the cars is at rest to start out with, this problem is pretty simple because all the momentum before the collision is in the moving car, m1. Momentum is mass times velocity, so it's m1 times v1. And the momentum of this, p equals 0. And after the collision, that momentum will be in the coupled cars moving together. So p final equals the total mass times the final velocity, so m1 plus m2 times v final. So now let's give the cars some masses. Say we give them the same mass at 2 kilograms, and we give the first car a speed of, say, 12 meters per second. Then we work it out to get a total initial momentum of 2 kilograms times 12 meters per second equals 24 kilogram-meters per second. And that equals our final momentum, which is m1 plus m2, so that's 4 kilograms, times v final. And in order to get 24, v final has to be 6 meters per second. So my v final equals 6 meters per second. So 6 meters per second, we ended up with half of the initial velocity, which makes sense because we said the two cars have equal masses. And we only had one of the masses moving at the start. So now you've got double the moving mass at the end, so they must be going half of the original speed if momentum is conserved. And that's our basic inelastic collision problem solved using conservation of momentum. But what about conservation of energy? Is kinetic energy conserved in our system here? Surprisingly no. Let's see why by working out the math. Let's calculate the energies before and after the collision. Kinetic energy initial is just going to be 1/2 mv squared. Before the collision, we only have the first car moving. So the total energy is 1/2 m1 times v1 squared, which is going to be 1/2 times 2 kilograms times 12 meters per second squared, which works out to 144 joules. After the collision, there's-- so our final kinetic energy equals 1/2 mv squared. But my final mass is both masses together, which equals 4 kilograms. So this equals 1/2 times 4 kilograms, and then my final velocity squared, so my 6 meters per second squared. And that equals 72 joules. So we lost kinetic energy. So what happened to the energy? Well, in an inelastic collision, when the colliding objects stick, some of the original kinetic energy is lost. Some of it escapes as heat. Oftentimes it goes into the objects mechanical deformation, like if you throw an egg against the wall and it deforms, or a ping- pong ball hits some fruit and it deforms. The energy isn't lost to the universe, but it is no longer in the system of the two things that collided, so we can't easily measure it. So when you have a collision where things stick together afterwards, you can't use conservation of energy to solve those problems. And that's where the word "inelastic" comes from. The objects that collide deform, and they don't elastically return to their original state. So done with inelastic. Let's move on to talking about elastic collisions. A lot of times, colliding objects do keep their shape, or at least they return to it after a little bit of jiggling. In these cases, very little to no energy is lost in the collision. We call these "elastic collisions." Although most collisions, even the ones that you talk about in your physics class as being elastic, are not perfectly elastic. Think about that water balloon that didn't break. You can see it wobbling. So some energy was lost to deformation or to heat. And there is almost always some energy loss in a collision. But we said very little energy is lost. So it's often a good enough approximation for most problems. Cool, moving on. So, the classic example for elastic problems is pool, like the game. I think pool is actually a controversial game because it's the source of a historical inaccuracy in one of Shakespeare's plays. In Antony and Cleopatra, Cleopatra exclaims, "Let's to billiards." But billiards wasn't a thing in the real Cleopatra's time, although it was popular in Shakespeare's time. For shame, William. Anyways, we're playing pool. I have two pool balls, and I shoot one towards the other. And they collide, and I want to know how fast they're going after the collision. Let's say that the two balls have a mass m. And the first one has some initial velocity v1. Let's say it's 2 meters per second. And the second one is sitting still at the start. So v2 equals zero. And then the balls hit.

(Describer) She draws them after that happens.

And the first ball will have some final velocity that we don't know yet. And we're going to call it v1 prime to not get it confused with the original v1. And then the second ball also have some mystery final velocity, v2 prime. We want to know what the final velocities are. Let's use conservation of momentum first. The momentum of both balls before equals the momentum of both balls after. So we write that mathematically like this. My initial momentum equals my final momentum. And the velocity here is zero, so my initial momentum is just this one, m1v1. m1 is just m, so it's mv1. My final momentum is the sum of these two, so it's going to be m1, which is just m, so mv1 prime plus m2, which is just m, v2 prime. And then since the masses are the same across the board, then every mass term of the entire equation is the same. So we can even divide out all of the terms by m. And we get a super simplified version of this, which is v1 equals v1 prime plus v2 prime. This only works out this way because the two balls have identical masses. Otherwise, we wouldn't be able to cancel out the masses. Now, we know v1, but we don't know v1 prime or v2 prime. So we have two unknowns but only one equation. But the elastic collision conserves energy, so we can use that idea to get more insight mathematically. Before the collision, the second ball didn't have any kinetic energy because it was sitting still. So the total initial kinetic energy is going to be the kinetic energy of the first ball. So that initial is 1/2 mv squared of this ball. So it's going to 1/2-- the mass of m1 is m-- 1/2 mv1 squared. And conservation of energy, that's going to equal my final energy, which is going to be the sum of the kinetic energy of these two, which is going to be 1/2 mv1 prime squared plus 1/2 m2, which is m, v2 prime squared. So here's our conservation of energy equation. And because we have a "1/2" and an "m" in every single term, we can just divide it out to simplify and get V1 squared equals V1 prime squared plus v2 prime squared. Again, this works because the pool balls have identical masses. Now we have two equations, the conservation of momentum one and the conservation of energy one, and two unknowns, the final velocities of the two balls. Great. But, ugh, this is some nasty algebra. And you can imagine that if the masses were not equal, it would be even nastier. So I'm going to show you a trick. I'm not going to do the derivation here, even though it would have been awesome. You can look it up in almost any physics book or somewhere online.

(describer) Title: Search elastic collision relative velocity.

(Describer) (no standard option) Title: Search Elastic Collision Relative Velocity.

And you should because it's a pretty cool derivation. But I'm just going to tell you. When any two objects collide elastically with each other and the energy is conserved, then the difference in the two velocities before the collision equals the negative of the difference between the two velocities after the equation. This is so cool. Mathematically that is, v1 minus v2 equals negative v1 prime minus v2 prime. The difference of the initial velocities is the negative of the difference between the final velocities. And you should look up this derivation. Whatever the masses are, the relative speeds of the objects will always be the same, except for the minus sign. This is really nice because it gives us a much simpler equation to solve. It's basically conservation of energy, but it's packaged in a much nicer box. So this new simple equation along with our conservation of momentum equation lets us solve for the two unknowns. So we're going to plug v1 prime plus v2 prime. And that equals minus-- Remember we say minus v2, but v2, as you recall, was zero. So I'm just going to go ahead and get rid of it. And then we say that that's equal. I'm going to distribute this negative to negative v1 prime plus v2 prime. Now bring this over, this negative, and I get 2v1 prime equals-- and then I'm going to subtract on both sides for the v2 prime. I'm going to subtract this from over here, and I get a 0. So it turns out v1 prime equals 0. And then I can plug that back in, and I get v1-- v2 again is 0-- equals minus-- v1 is 0. So I get 0 minus v2 prime equals v2 prime. So that means v2 prime is equal to my initial vi, which was 2 meters per second. So v2 prime equals 2 meters per second. Well, this is cool because the math in our answer is telling us something interesting. Ball one completely stops, and ball two moves on with the velocity that ball one had. In this case, ball one ended up transferring all of its momentum and kinetic energy to ball two. The balls switched velocities. And look at this equation. If we stop and think about it for a minute, it turns out that this situation of having one ball stopped and the other one moving is just a special case of this general case of two balls with the same mass colliding. It turns out no matter what the initial speeds of the two balls, if they have the same mass, they just swap speeds. That's very cool. And that is Newton's cradle.

(Describer) She puts it back on her table.

[grunts]

When you drop the ball on one end, it hits the second ball and transfers its momentum and energy to it. It has the same mass. And then the second ball transfers that momentum and energy to the third ball, and so on until the last ball shoots up with the same momentum and energy that was transferred to it from all the other balls. So it should actually reach about the same height.

(Describer) She holds a hand out to measure. The end balls swing the same height.

So it should come to about here. Yep, so cool. So these two collisions, the inelastic toy trains and the elastic pool balls, had very different outcomes. But remember this important thing that they have in common: the total momentum in the system before and after the collision is the same. And actually, there's another deeper way of thinking about this. It's a little tricky to wrap your brain around, but so great. What if instead of thinking about the sum of momenta, we were thinking about the center of mass of the system? The center of mass of the system actually keeps moving with the same velocity before and after. Here's a fun party trick to visualize what the center of mass is. Hold out a yardstick or a pipe with each end of it

(Describer) She does.

balanced on one of the fingers of each hand. And then move the fingers inward along the pipe. I promise you no matter how you do this, you cannot do it without your fingers meeting in the middle,

(Describer) Hers do.

always, no matter how you try.

[light blowing sound]

(Describer) She does it again.

[giggling]

It's easier with a yardstick because--maybe I'll try it like this.

(Describer) She curls her fingers under the pipe.

Always in the middle. Try it for yourself. You are ending up in the center of mass. Why is the middle of the pipe its center of mass? That's because the pipe is homogeneous. It's got the same material, and it's the same mass on either side of the center of mass. If you did this with an object that's heavier on one side than the other, like a broom, then you would still find the center of mass by doing this, but it won't be in the middle. All systems have a center of mass, even when not all of the parts are connected. So with the balls and the trains, the system has the center of mass too. Because the balls and train cars have the same mass in our examples, then the center of mass is just exactly in the middle of the two objects. Let's take the train, for example. I'm just going to draw it. We had two trains. Train one was moving at 12 meters per second,

(Describer) Right.

and the second car was stopped over here. So the center of mass was halfway in between. And that was moving to the right at half the speed of this one at 6 meters per second.

(Describer) The first.

So the halfway distance is going to shrink at half of the speed of the full distance. But after a collision when the two trains stuck together, you may recall they were going 6 meters per second, which is the same as the center of mass of our system. So during this whole process, the center of mass of the system kept moving at the same speed and never stopped. And that's always true. And that is just a fundamentally awesome idea. And this is inertia. As long as there are no external forces on a system to change the velocity of its center of mass, then the system will keep going with the same motion, even if the pieces within it are moving around. And you can see examples of that all over the place. Like Romeo and Juliet, say they're having a romantic stargazing evening and they suddenly see a bright dot in the sky, which turns out to be an exploding supernova many, many light-years away from Verona. That supernova is shooting its innards every which way, which is like a reverse collision playing backwards. But despite the fact that the pieces are moving in different directions, the center of mass of all the pieces of the star are going to keep going along whatever its original trajectory was, maybe orbiting another star, probably orbiting the center of mass of its galaxy as well. And so far, the examples we've worked through here were objects with identical masses, the two train cars, the pool balls. But what happens if we have balls with different masses? Here we have multiple collisions going on from multiple balls. But let's reduce it to a simpler situation with just two objects, like a basketball and a tennis ball. Both fall together at the same rate because all objects near Earth accelerate the same rate if we ignore air resistance. And the basketball hits the ground first because it's below the tennis ball. So it hits the ground and compresses and rebounds before the tennis ball does. And it actually starts traveling up into a tennis ball. And that's a collision. The basketball will be traveling upward into the tennis ball at about the same speed as it did when it hit the ground, which is also about the same as the speed of the tennis ball still traveling downward at the moment of the collision. So at that point, this becomes a conservation of energy problem. There's a huge mass difference between the two balls. And when you work it out, you get that the velocity of the basketball is almost unchanged, but the tennis ball has a much higher velocity after the collision. And it goes really high. And check this out. I actually filled a basketball with silicone so it weighs a lot more than a regular basketball. And it trips people out because you pick it up, you think it weighs the same as regular basketball, and it's very confusing. If I drop that basketball with the tennis ball, look how high it goes. Because with a super large mass, the silicone basketball has a ton of kinetic energy to pass onto the tennis ball. You can think of this whole thing like two people jumping on a trampoline. If Romeo starts jumping before Juliet, he is stretching the trampoline already preparing it for her. So that stores some energy in the stretched trampoline in the form of spring energy, potential energy. And then when Juliet jumps and lands on the trampoline, she'll be rocketed up by that extra stored potential energy. So with the basketball and the tennis ball, you're storing some energy by compressing the basketball when it hits the ground. And that gets transferred into the tennis ball and shoots it higher into the air. And that's today's lesson on collisions.

(Describer) She moves the paper away slowly.

(Describer) She tosses the marker.

Oh, almost got it. When they ask what you learned today on YouTube, here are your two key takeaways. Number one, conservation of momentum is the tool for all types of collisions. And number two, conservation of energy only holds for elastic collisions, but use the negative relative velocity trick when you can. And because practicing solving problems is a key part of learning physics, here are all the problems from today's lesson. Go work them again. And if you dare, search more conservation of momentum and collision problems online.

(describer) Problem one: Quiz!

(Describer) Problem 1: Quiz! Are the following collisions elastic or inelastic? Asteroid hitting Earth? Spitball smacking into your friend? Pool balls hitting each other? Water balloon hitting your face? Newton's Cradle? Bumper cars colliding? A car crash? Subatomic particles colliding? Black holes colliding? Problem 2: A toy train of mass 2 kilograms, traveling at a speed of 20 meters per second, collides with an identical train at rest. They stick together. What is the new speed of the trains? How much energy was lost to heat in that inelastic collision? Problem 3: A pool ball traveling at 2 meters per second directly hits an identical but stationary pool ball. What are the velocities of each ball after the collision?

Are the following collisions elastic or inelastic? Asteroid hitting Earth? Spitball smacking into your friend? Pool balls hitting each other? Water balloon hitting your face? Newton's cradle? Bumper cars colliding? A car crash? Subatomic particles colliding? Black holes colliding? Problem two: A toy train of mass 2 kilograms, traveling at a speed of 12 meters per second, collides with an identical train at rest. They stick together. What is the new speed of the trains? How much energy was lost to heat in that inelastic collision? Problem three: A pool ball traveling at 2 meters per second directly hits an identical but stationary pool ball. What are the velocities of each ball after the collision? Before we wrap it up, I just want to point out one more important thing about collisions. So far, the examples we've gone through have been for motion and conservation of momentum in one dimension along one direction. But all these ideas carry over into two and three dimensions as well. For example, when you're playing pool, you typically don't just play in one line. You might hit a couple pool balls, so they speed away on diagonal paths relative to your initial hit. And sometimes, they even jump off the table if you play like I do. But if you look at the balls' velocities in the x and the y direction separately, you will see that both are conserved in those respective directions, and in the z direction. This is partly how physicists get information about particles and their masses, and their other properties in particle accelerators like the Large Hadron Collider at CERN in Switzerland, the place where they discovered the Higgs boson. If a particle with no momentum in the y direction, just not moving up and down, splits into pieces and goes flying off with some y momentum, we know that the total y momentum of all the pieces has to sum to zero. So physicists can use that information about the masses of the particles that flew apart. Such a simple idea gives amazing and really deep insight into the basic particles of our universe. And that's how we figure out masses of things like the pieces of a proton. That's insane. And if you go on and take a class in particle physics, you can learn more about this awesome type of physics and maybe do research in it one day. I do hope you continue on with physics. And now a message for you from a special guest. Hello. My name is Katie Mack, and I'm an astrophysicist specializing in cosmology, which is the study of the universe from beginning to end, how it evolves, how it changes over time, all of the weird stuff in it. I am often asked how one becomes an astrophysicist. And one of the things I say is to learn as much as you can. Follow your curiosity. And it's awesome that you are doing that. And you were going to have so much fun discovering the mysteries of the universe. And I can't wait to find out what you find out.

(Describer) Titles: Executive Producer/Host/Writer: Dianna Cowern Lead Writer/Course Designer: Jeff Brock Producer: Laura Chernikoff Video Editor: Kaitlyn Ali Curriculum Consultants: Lucy Brock and Samantha Ward Accessibility provided by the US Department of Education.

Accessibility provided by the U.S. Department of Education.