Section 21.8

Dirac Delta Function

A force large enough, fast enough, that it happens in an instant. The mathematical model for a "kick".

1

Introduction

Sometimes forces act over such a short time that we don't care about the shape of the force, only the total Impulse (Change in Momentum).

Examples: Hammer hitting a bell, lightning strike, golf club hitting a ball.

2

Variables

Definition

The Dirac Delta is defined by:

  • for .
  • . (Unit Impulse).

Sifting Property:

.

3

Visual: Limit of a Pulse

Interactive: Shrinking Width

Physically, it's a very tall, very thin rectangle with area 1.

4

The Transform

Using the sifting property with :

Special Case: If , .

Recall . The Delta function is even "stronger" than a constant.

5

Worked Examples

Example 1: Impulse Response

Solve , .

1. Transform:

.

.

2. Inverse:

Base function .

Shift by .

.

3. Simplify:

.

.

Example 2: Velocity Jump

Interpret the solution from Example 1.

Before , .

At , the system receives a "kick".

The solution instantly starts oscillating.

Check "Velocity" just after :

.

.

The impulse unit 1 changed the velocity from 0 to 1 instantaneously.

6

Practice Quiz

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