Impulse (J) is the product of the average net force applied to an object and the time interval over which that force acts:
Impulse is measured in N·s (Newton-seconds), which is identical to kg·m/s — the same units as momentum. This is not a coincidence. It is the direct consequence of the impulse-momentum theorem.
Like momentum, impulse is a vector. Its direction is the same as the direction of the net force. A rightward force produces rightward impulse; a leftward force produces leftward impulse.
The impulse delivered to an object exactly equals its change in momentum:
This is the most important equation in Lesson 4.2. It directly links what a force does (impulse) to how motion changes (momentum change). Use it whenever you know — or want — force, time, mass, or velocity information at two points in time.
Set the force and contact time to deliver an impulse. Watch how it changes the object's momentum and final velocity directly.
Same impulse, longer time = slower change. Try halving F and doubling Dt — same J, same Dp, same final velocity. This is why airbag contact time matters.
A 0.4 kg soccer ball is kicked from rest and reaches 15 m/s. The foot is in contact for 0.08 s. Find the impulse and the average force exerted.
Newton's Second Law (F_net = ma) is not a separate law from the impulse-momentum theorem — it is a direct mathematical consequence of it. Here is the derivation:
The more general form — valid even when mass changes — is F_net = Dp/Dt. This is actually how Newton originally stated his second law. The familiar F = ma is the special case for constant mass.
The impulse-momentum theorem has two powerful graphical forms that the AP exam tests almost every year. Know both fluently.
The area under a Force vs. time graph equals the impulse delivered. For constant force: area = rectangle = F * Dt. For variable force: count grid squares, use geometry (triangles, trapezoids), or estimate.
The slope of a momentum vs. time graph equals the net force at that instant. A steep slope means large force. A horizontal line means zero net force — momentum is constant. A curved p vs. t means changing force.
A Force vs. time graph shows a triangular pulse: force rises linearly from 0 N at t = 0 to 120 N at t = 0.3 s, then drops back to 0 N at t = 0.6 s. A 3 kg object starts at rest. Find its final speed.
The most important real-world application of the impulse-momentum theorem is the trade-off between force and time. For a fixed change in momentum, increasing contact time decreases the average force — and vice versa.
A 70 kg driver hits the steering wheel and decelerates from 15 m/s to rest. Without an airbag the stop takes 0.02 s. With an airbag it takes 0.25 s. Find the average force in each case.