d'Arbeloff Interactive Math Project

  Football Trajectory

The nonlinear vector-valued differential equation ma = -mg j - b|v|v controls the flight of a projectile subject to quadratic air resistance. Here m is the mass, g is the force of gravity, b is the constant of air resistance, j is a unit vector directed upwards, v is the velocty vector, |v| is its magnitude, and a is the acceleration vector.

Sliders at bottom select values of m and b , as well as the initial speed v0 and angle up from horizontal, θ0 .
The initial conditions can also be selected by a mouse click on the graphing window.

Initiate an animation by pressing [throw] at lower right, stop it by pressing [stop], and resume it by pressing [continue].

In the graphing window, the flight path is depicted in white. The projectile is represented by a red dot, and the animation leaves behind a trail of red dots at equal time increments. The initial velocity vector is indicated by a blue arrow, the force of gravity by an orange arrow, and the resistant force by a yellow arrow.

A cursor rollover produces crosshairs and a readout of the x and y coordinates.

 

© 2007 P. Dourmashkin, H. Miller and J.-M. Claus