A step function is 0 before a moment and 1 after it. It models a switch closing or a load suddenly applied.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Why they are awkward otherwise
A discontinuous forcing term breaks the usual guessing methods. The Laplace transform handles it without special cases.
Impulses
An impulse is a very large force acting for a very short time. A hammer blow or a kick is the physical picture.
What an impulse does
It changes the velocity instantly while leaving the position where it was. The state jumps in one coordinate only.
After the event
Once the impulse has passed, the system evolves freely again from its new state. Nothing else about the equation has changed.
Why engineers use them
The response to an impulse characterises a whole system. Knowing it lets you predict the response to any input at all.
Switches that flip
The unit step function is zero before a time and one after. It models a switch closing or a force being applied, and its transform is simple, which is the transform's main advantage.
Piecewise forcing written compactly
Combinations of shifted step functions express any piecewise-constant input as a single formula. That is what lets the transform handle inputs that would otherwise need the problem split into intervals.
Forces that strike
The delta function models an idealised instantaneous impulse — a hammer blow, a sudden voltage spike. It is not a function in the ordinary sense, and it is defined by how it behaves inside an integral.
The impulse response characterises the system
The response to a delta input determines the response to any input, by convolution. That is why engineers measure impulse responses: one experiment characterises the whole system.
Step 2: Try It Yourself
Tap and try it out.
- Point(1, 1)
Step 3: Watch an Example
One step at a time.
Watch Priya Track an Impulse
Priya has a mass at rest at position 0 that receives an impulse giving it a velocity of 5.
- Step 1
She notes the position immediately before the impulse, which is 0.
Step 4: Your Turn
Practice makes it stick.
The Kick
Problem 1 of 2
A mass at rest at position 3 receives an impulse. What is its position immediately after?
The Switch
Problem 2 of 2
A step function turning on at t = 4. What is its value at t = 2?
Sudden Changes
1 of 8
A step function turning on at t = 3. What is its value at t = 5?
2 of 8
A step function turning on at t = 3. What is its value at t = 1?
3 of 8
An impulse applied to a mass. Does the position jump? 1 yes, 0 no.
4 of 8
An impulse applied to a mass. Does the velocity jump? 1 yes, 0 no.
5 of 8
A mass at rest receives an impulse giving velocity 8. What is its velocity immediately after?
6 of 8
A mass moving at 2 receives an impulse adding 6 to its velocity. What is the new velocity?
7 of 8
Sort each quantity by whether an impulse changes it instantly.
Tap something to move it.
- Empty
- Empty
8 of 8
A step function turning on at t = 0. What is its value at t = 10?
Step 5: Quick Check
Show what you know.
Question 1 of 2
A mass moving at 3 receives an impulse adding 4 to its velocity. What is the new velocity?
Question 2 of 2
What does an impulse change instantly?
What You Learned
- A step function models a switch flipping, and the Laplace transform handles it without special cases.
- An impulse is a large force over a vanishing time: it changes velocity instantly, not position.
- After the event the system evolves freely again from its new state.