Unit Step and Impulse Functions

Differential Equations · Axiom Academy

A mass on a spring is sitting still. Then something happens to it — a switch flips, or a hammer strikes. Two functions model those two kinds of "something." A 1 kg mass hangs on a spring ( k = 4 N/m ) with light damping ( c = 0.4 N·s/m ). It starts at rest. Then, at some moment, a force acts on it — sustained or instantaneous . Same system, two very different inputs. The Heaviside (unit step) function Drag when the force turns on, hit ▶ Run, and watch the mass respond — the force switches on instantly and then holds steady forever after. u(t-a) is 0 right up until t=a , then jumps to 1 and stays there — a perfect on/off switch. The Dirac delta (impulse) function Now hit the SAME mass with a hammer instead — an instantaneous strike, not a sustained push. Drag when the strike lands and watch how differently the system responds. is an idealized force: infinite for an instant, zero everywhere else, delivering a finite total impulse — . Same system, two inputs, two very different stories Overlay both responses for the identical mass-spring-damper system, both triggered at the same moment. Toggle each on and off to see what each input leaves behind.

This is the written version of the interactive lesson above. See the full Differential Equations course.