Step Functions and Impulses

Switches that flip and forces that strike.

12 problems · about 23 minutes · F-IF.C.7, F-BF.B.3

What this lesson teaches

The student models discontinuous and impulsive forcing and describes the response.

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.

Straightforward practice. Get the method working first.

5 problems

A mass moving at 3 receives an impulse adding 4 to its velocity. What is the new velocity?
Answer 7
Why 7.
What does an impulse change instantly?
Answer The velocity, while the position stays where it was.
Why Velocity only.
A step function turning on at t = 3. What is its value at t = 5?
Answer 1
Why After the switch.
A step function turning on at t = 3. What is its value at t = 1?
Answer 0
Why Before the switch.
An impulse applied to a mass. Does the position jump? 1 yes, 0 no.
Answer 0
Why No time passes.

The same ideas with more to keep track of.

3 problems

An impulse applied to a mass. Does the velocity jump? 1 yes, 0 no.
Answer 1
Why That is what an impulse delivers.
A mass at rest receives an impulse giving velocity 8. What is its velocity immediately after?
Answer 8
Why 0 plus 8.
A mass moving at 2 receives an impulse adding 6 to its velocity. What is the new velocity?
Answer 8
Why 2 + 6.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Sort each quantity by whether an impulse changes it instantly.
Answer Changes instantly: Velocity, Momentum · Stays the same through the impulse: Position, The location of the mass
Why Momentum is mass times velocity.
A step function turning on at t = 0. What is its value at t = 10?
Answer 1
Why Long after the switch.
The Kick: A mass at rest at position 3 receives an impulse. What is its position immediately after?
Answer 3
Why 3.
The Switch: A step function turning on at t = 4. What is its value at t = 2?
Answer 0
Why 0.