Desmos Flywheel Projectile Calculator

By David, Nalin, Fernando, and Anda

When building a flywheel launcher, understanding the physics is essential. Our Desmos-based trajectory simulator helps us model and test different launch scenarios using kinematic equations.

At the core of the simulator is the projectile motion equation: y = x·tan(θ) - (g·x²)/(2(v·cos(θ))²). This combines horizontal and vertical motion into one formula. The x·tan(θ) term represents the initial upward direction of the launch, the second term accounts for gravity pulling the projectile down, and Y0 adds the effect of any launch height. We use this same equation twice with different velocity inputs to compare two launch scenarios.

The purple trajectory can use any initial velocity inputted, giving us the textbook projectile motion path. The red trajectory plugs in a velocity calculated from our flywheel system, where V_tangent = F_rpm·(2π·F_radius) This equation converts rotational speed (in revolutions per minute) into linear velocity at the flywheel’s surface. It models how a spinning flywheel actually launches the projectile. Our simulator also accounts for gravity (g=9.81 m/s2), launch height, motor efficiency, and gear ratios, giving us a complete picture of how the system behaves under realistic conditions.

To visualize accuracy, the simulator includes a goal box that defines our target area. This box is created using boundary equations that set the top and bottom y-limits between specific x positions, forming a clear rectangular “hit zone.” It’s immediately obvious whether the trajectory passes through the goal or misses.

This saves a lot of time compared to building and rebuilding physical prototypes. We can test dozens of configurations in minutes and understand which parameters actually matter for hitting our goal. The visual feedback makes it clear when adjustments bring the trajectory closer to passing through the goal box or when we need to try a different approach. The combination of physics modeling and visual feedback makes it much easier to understand what's happening and make informed decisions about our build.

If you want to check out the calculator for youself, here it is: Desmos Trajectory Calculator