Here's a problem that illustrates an important technique in space exploration. I have a heavy ball and a light ball, and I'm going to drop them like this. After the heavy ball bounces, but before the light one does, what is the center of mass doing? Then, when the two balls collide, what will happen? The heavy ball bounces up and meets the light ball coming down. Well you know what's going to happen. What happens when a light ball strikes a heavy bat going in the opposite direction? But let's see. Why is that important to spaceflight? Think of the Voyager spacecraft, the ones that are leaving the solar system. Where did they get their mechanical energy? Only some of it came from rockets. They also had a flyby, a sort of elastic collision with Jupiter: Jupiter, Voyager. Now Voyager didn't actually hit Jupiter because Jupiter's surface isn't springy. But, actually, it was an elastic interaction. Gravity and spring forces are both conservative, so the effect is similar. The two objects exchanged impulse, and Jupiter gave a minuscule fraction of its kinetic energy to Voyager, and helped it on its way. More recently, the Messenger spacecraft had an Earth flyby. This time, not to gain energy, but to lose it in a rear-end collision. More about orbits and spacecraft next week. Spacecraft planet interactions are elastic, but ball bounces are not. The ball is springing, but the deformations are not completely reversible, so some mechanical energy is lost in each bounce. These are inelastic conditions, but not completely inelastic. Here's a collision that is completely inelastic. The ball lodges in the pendulum and they travel together. Remember, not all kinetic energy is lost, because the center of mass kinetic energy can't be lost without external forces. This is called a ballistic pendulum. We can use the height of the rise of the pendulum to measure the speed of the ball. Let's analyze this one together. First, consider the collision. What, if anything, is conserved? And what's the relation between the speed of the ball before the collision and the speed of the pendulum after? Well done. Now onto the next phase, as the pendulum swings up. What can we say here? Your job is to find the ball speed, V, as a function of H, the height of the pendulum swing. Good. You've shown us how to work backwards to find the speed of the ball from the height of the swing. The reason why we've included this example is to show how you can divide a complicated process into different phases, and then to think which conditions apply for each. In this experiment, neither momentum nor mechanical energy was conserved over all, but each one of them was conserved during one of the phases. Let's finish this section with another demonstration in which neither momentum nor mechanical energy was conserved overall. Obviously, there is a collision here. Following the collision, my chest compresses. The compression is a bit like that of a spring. I'm glad to report that, after the compression, it comes back to normal. Further, we did this experiment to measure the spring constant of my chest. So, after the collision, I can use conservation of mechanical energy to get a pretty rough estimate of the maximum compression of my chest. Obviously, I did this analysis carefully before I did the demonstration. We'll give you the details in the quiz. Here's an interesting point. Given that the results of this calculation are very important for my health, you might wonder why I was happy to use some pretty crude approximations. The reason is, I wanted a reliable answer, not an exact one. If the answer is several millimeters or less, I'm happy to do it. If it's several centimeters or more, then I obviously don't do the demonstration. But what is important is that I really understand the principles. At UNSW, I often teach engineering students. It's important for them to think that, soon, people's lives may depend on their calculations. A thorough understanding is really important, not just substituting into an equation. Let's practice that understanding in the quiz, and then there's the test. Next week, we'll be back with gravity. Planets, stars, black holes, dark matter. It's going to be fun. See you then.