Particle collisions

(metal clanging) - We finished going over the first of three things you need to create a particle simulator. We have formulas that describe how particles will move based on the laws of physics. In this video, we'll develop formulas that describe how particles will behave during collisions. If velocity is perpendicular to the floor at the moment of collision, and the collision is perfect, then the velocity reverses. In the real world, its direction reverses, and the magnitude is slightly reduced due to friction. As we introduced in lesson one, we can model the frictional losses by multiplying by a factor that measures the elasticity of the collision. So if V is the downward velocity before the collision with the floor, then the velocity, called V prime, after the collision is given by V prime is equal to negative E times V. Here E is the elasticity, a number between zero and one, and the minus sign indicates the direction reverses after the collision. If E equals one, the collision is perfect, and no energy is lost. If E is less than one, then some energy will be lost, but what if the velocity makes an angle with the floor. Let's observe what happens on video. Notice that the ball bounces just like a light ray reflects off a mirror. That is, if the incoming velocity V makes an angle Theta with the floor, then the outgoing velocity called V prime makes the same angle. To compute V prime from V, let's write V as the sum of two vectors, a velocity parallel to the floor, V parallel, and the velocity perpendicular to the floor, V perpendicular. The only force during the collision is perpendicular to the floor because the floor pushes up on the ball. So the parallel component won't change, and the perpendicular component, as before, will be reversed, meaning that V prime is equal V parallel minus V perpendicular. Adding elasticity into the mix, we get this equation. Cool, we can use the idea of writing the velocity as the sum of parallel and perpendicular components to study the case when two particles I and J of the same mass collide. Let's draw a picture to make this clear. The line I, J from the center of particle I to the center of particle J plays the role of the perpendicular to the floor. (bell dinging) So we write V, I as a sum of two vectors, as shown here. We can do the same for V, J. The only force acting on the particles during the collision is along the line I, J. As before, since no force is acting in the parallel direction, the velocities in the parallel directions won't change. To figure out exactly what happens in the perpendicular direction requires using more advanced topics, namely, conservation of energy and momentum. (chiming) If we apply those concepts and assume that the particles have the same mass, we find that particle I gets J's perpendicular velocity, and vice versa. That is, they swap perpendicular velocities. (bell chiming) That means after the collision, the velocities V prime I and V prime J are given by these equations. That's it for particle collisions. In the next exercise, you'll have a chance to review these equations, which describe collisions. (cheerful music)