Join me in this week’s STEM video, **the Geometry of Catapults**, as I take you on a **math adventure** to discover amazing insights into how our catapults work, and what we can do to tune the distance or range that our projectiles reach.

By looking at the **transfer of Potential Energy into Kinetic Energy**, we discover that our projectile velocity will increase as the spring constant of our rubber band **increases**, as the mass of our projectile **decreases**, and as the elongation of our rubber band **increases**.

Next we look at the effect of **launch angle** on our projectile’s trajectory and **total distance traveled**. We learn that the launch angle, alpha, controls how much velocity goes into a **horizontal** component of velocity versus a **vertical** component of velocity.

We learn that both have an affect on distance traveled, because the horizontal distance traveled, x, is equal to the horizontal component of velocity times the time to fall… which is related to the vertical component of velocity.

We can solve for the time to fall using the quadratic equation in y as a function of the vertical component of velocity and the initial height, y0, of the projectile. We can increase the time to fall by increasing the **vertical component of velocity** *or* the **initial height, y0**, of our projectile at launch. Here’s my crib sheet of key equations:

But if our question is how to maximize X (the horizontal distance traveled), what’s the optimum launch angle?

Good question! That’s for you to test. I take you through a few examples, using the velocity I measured from one of my catapult tests using video shot against a gridded background.

For a given initial vertical offset, y0, and total velocity, v=129 inches per second, here’s what the effect of different launch angles looks like: