4.2 Change in Momentum and Impulse

Impulse-Momentum Theorem

Impulse (J) equals the change in momentum: J = Δp = m·Δv. Impulse is also defined as the integral of force over time: J = ∫F dt. Since force and momentum are vectors, impulse is a vector too, pointing in the same direction as the net force applied over that time interval.

Force–Time Graphs

When force varies with time, impulse equals the area under the F-t curve. For constant force, J = F·Δt. This lets you calculate impulse from graphs without knowing the exact force function, using geometry (rectangles, triangles) or calculus (integration) depending on the shape of the curve.

Real-World Applications

Increasing the time of collision (Δt) while keeping impulse constant reduces the average force experienced — this is why airbags, padding, and bending your knees when landing reduce injury. Since J = FΔt is fixed by the momentum change, a longer Δt means a smaller average F.

Key terms

Impulse (J)

The change in momentum of an object; J = Δp = ∫F dt, a vector quantity.

Impulse-Momentum Theorem

States that the net impulse on an object equals its change in momentum: J = Δp = mΔv.

Impulse from constant force

J = F·Δt when force is constant over the time interval.

Impulse on F-t graph

Equal to the area under the force vs. time curve.

Why extending collision time helps

Since impulse is fixed by Δp, increasing Δt decreases the average force (F = J/Δt), reducing injury or damage.

Try an AP-style question

F(t) = 10t N acts from t = 0 to t = 2 s. What impulse does it deliver?

1. 10 N·s
2. 20 N·s
3. 40 N·s
4. 5 N·s