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- The final step to complete our particle simulator is to develop a method to track particle motions forward in time. That's how our computer program will animate them. One frame at a time. We said earlier that if you know the equation of motion of each particle, then you can compute the velocities and positions from it. But how exactly? To answer that question, let's go back to the idea of velocity as the slope of the position versus time curve. Put two values of time, t one and t two, close together, and let p one be the position at time t one and let p two be the position at time t two. The slope of the line l, shown here, is a good approximation of the velocity v one at time t one. The closer that t two gets to t one, the better the approximation. As an equation, the slope of l, that is the velocity v one, is given by the change in position divided by the change in time. If we know the position and velocity at time t one, then we can compute the position in time t two by rearranging this equation to solve for p two. Great. So knowing the particle's position and velocity at time t one, we can compute the position at time t two using this formula. But how do we get the velocity at time t two? Well, if we know the equation of motion, then we can compute the acceleration at time t two. For instance, if the particle is just being acted on by gravity, then the acceleration is constant, and is given by the gravitation constant g. We also know that the acceleration is the slope of the velocity versus time curve, meaning gravity equals a change in velocity divided by the change in time. And we can solve this for v two. Now that we know p two and v two, we can repeat this process to compute p three and v three and so on, for as long as we like. Let's do an example. Suppose that at the start of the simulation, we set our time parameter t to 0, our particle is at point p one with velocity at v one, and the gravity vector g points down. To figure out where the particle will be at time t equals one half, we use the equation p two is equal to v one times the quantity t two minus t one plus p one, where t one is equal to zero and t two is equal to one half. So, p two is equal to one half v one plus p one. Meaning that p two is halfway between v one's tail and head. And to figure out v two, we use v two is equal to g times the quantity t two minus t one plus v one. G here is the gravitational vector which points downward, and on Earth, has a magnitude of 9.8 meters per second squared. Again, t two minus t one is one half, so v two is equal to one half g plus v one. Great. Now we can compute the position and velocity at t equals one using the same formulas. Although this is rather tedious to do by hand, it is relatively easy to write a computer program to do these calculations for us. Like this. Congratulations. We now have all the parts we need to create a ping pong ball simulator. (chimes) In the final exercise, you can test your understanding of these concepts before moving on to create your own amazing particle simulations.