Step and Delta Function Motivation

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Section 16.2 Step and Delta Function Motivation

Consider a mass sliding along a frictionless surface. It collides, elastically, with a wall. The velocity of the mass as a function of time is shown in Figure 16.3 and its acceleration is shown in Figure 16.4.

Figure 16.3. The velocity of an idealized mass colliding with a wall.

Figure 16.4. The acceleration of an idealized mass colliding with a wall.

Clearly, this is a highly idealized situation. In the real world, the mass will slow down slightly due to a small amount of friction, the interaction with the wall will not be instantaneous, nor will it be completely elastic, and the rebound will be somewhat slower than the speed when the mass hits the wall, etc. But suppose these differences from ideal are very small compared to the motion itself and that we do not care about the details, only the overall qualitative behavior. Then, we can model the behavior with two new functions, the step function (defined in Section 16.3) and the delta function (defined in Section 16.4). These functions are really easy to work with in algebra and calculus settings. With a little bit of effort now, you will come to welcome them in your work.

The tricky part, of course, is what happens at the moment (labelled \(t_0\) in the figures) that the mass hits the wall. All of the complications of the motion are compressed into that one time. The velocity changes instantaneously from its initial value \(v_0\, \hat{x}\) to \(-v_0\, \hat{x}\text{.}\) This behavior will be modeled by the step function.

Even worse, is the acceleration. The acceleration is the derivative of the velocity with respect to time, so the acceleration is zero before the mass hits the wall and after the mass hits the wall. But what is the value of the acceleration at the moment the mass hits the wall? Clearly, the acceleration is infinite. The delta function is what allows us to specify exactly how big this infinity is, so that the velocity by the correct amount.