Projectile Motion and Why I’ll Never Do it Again
This year, I wanted to figure out how projectiles move in the air. The idea was to calculate the best shot parameters to always fire a projectile accurately. If successful, I would unlock the key to many of life’s mysteries, like automating our FRC robot’s shooter or how to always peg people in dodgeball :). Like anyone else, I assumed that it would be a simple problem, but it turned out to be one of the most complicated and difficult problems I encountered. What appears to be a simple physics problem would turn out to be an ongoing problem in the field of physics, a problem that stumped great Mathematicians like Newton or Euler. Let’s start simple.
First of all, we have to understand the physics behind what makes things move. Motion can be modeled through the equation with Newton’s 2nd Law, which will allow us to find out how much drag will affect a particle’s motion. There are two main types of drag: linear and quadratic. The equations related to drag are shown below.
Linear Drag is when an object experiences a force proportional to its current velocity.
Quadratic Drag is proportional to velocity squared.
The Drag coefficient describes how much drag an object will experience. You can find this drag coefficient experimentally.
Linear Drag
We will represent our motion in components. One in the x direction and one in the y direction. The net force in the x direction is only the force of drag. The net force in the y direction however is the force of drag and the force of gravity. We will substitute mass times acceleration for force in both equations. The equation for the drag forces are shown below.
If we rewrite the acceleration as dv/dt, and move around terms, we get a differential equation, which can be integrated to solve for velocity. We can repeat the process if we wish to get position from velocity. After deriving these equations, we can model our trajectory as a parametric equation dependent on t. I have created a desmos model to show the trajectory of the object. It includes equations for position and velocity that can be turned on as well.
Quadratic Drag
Linear drag seemed pretty simple right? Problem is that it isn’t that accurate for most objects. If you want to model something worthwhile, you need to use quadratic drag. No problem! If we just integrate twice, I get to be the best at dodgeball! Let’s integrate for velocity.
We create our equations for each component and substitute for v.
Do you notice anything strange about the equations?
If we try solving for velocity in the x direction, we have to solve for it in the y direction, which requires that we solve it for the x direction again, which… When this happens, you have a coupled set of differential equations. They can’t be solved analytically, and you’ll have to use some kind of computer program to approximate. Fortunately, I found a paper that made an analytical approximation for the trajectory. It does a pretty good job up till a certain point. I talk about it in the desmos model.
Well, that’s pretty sad. We can’t model motion accurately for the majority of objects. It get’s even more complicated depending on the mechanism used for shooting. Originally, I was planning on using this for our FRC shooter in this year’s challenges. The projectile we launch is a ball, and we launch it using a flywheel. Flywheels work by spinning at high speeds to launch projectiles. However, they generate a lot of this force from compression, and not their actual speed. This means that the ball would not exit the flywheel at a predictable velocity, unless we wanted to do a lot more math. Additionally, the flywheel wouldn’t touch each ball the exact same, and a small variation in its surface could ruin the model. After doing this project, I learned how complicated modeling the real world is. I had an assumption starting this project that anything can be modeled. The fact that it behaves in a certain way should be enough to find some math to characterize it right? This was an interesting project, and one of my first with the knowledge of Calculus; it was a nice implementation of what we learned, and I’m glad I did it.