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Why impulse equals the momentum change

A steady 6 N net force acts on a wagon for 2 s, and the wagon's momentum grows by 12 kg·m/s. The same force acts for 4 s instead, and the momentum grows by 24 kg·m/s. Exactly double.

Why does doubling the time double the momentum change?

The impulse-momentum theorem, FΔt = Δp, is a rearrangement, and the rearrangement is the reason.

Start from F = Δp/Δt.

F = Δp/Δt says the net force is the momentum delivered each second.

Multiply both sides by Δt: FΔt = Δp.

The same 6 N force, three durationstime actingmomentum delivered0.5 s3 kg·m/s2 s12 kg·m/s4 s24 kg·m/s
6 kg·m/s arrives each second, so the time sets the total.

In words: momentum delivered each second, times the number of seconds, is the whole momentum change.

The rearrangement, move by movemovelinestartF = Δp/Δtmultiply both sides by ΔtFΔt = Δpread it in wordsrate × seconds = whole change
The rearrangement says: delivery rate times seconds is the whole delivery.

The 6 N force delivers 6 kg·m/s of momentum every second.

Keep it acting for 2 s, and 12 kg·m/s arrives. Keep it acting for 4 s, and 24 kg·m/s arrives.

A steady 5 N net force changes a cart's momentum by 15 kg·m/s in 3 s. The same force acts for 6 s instead. Enter the new momentum change in kg·m/s.

A fixed net force acts on a cart. Doubling the time it acts doubles the cart's momentum change. Why?

Twice the seconds means twice the momentum delivered. That is why the impulse FΔt equals the momentum change Δp.

A steady 6 N net force acts on the wagon for 4 s. Calculate its momentum change.

Write down the values:

F = 6 N, Δt = 4 s, Δp = ?

Write down the equation:

FΔt = Δp

Substitute in the values, and calculate:

Δp = 6 × 4 = 24

The wagon's momentum changes by 24 kg·m/s: 6 kg·m/s for each of the 4 s.

The same 6 N net force acts for only 0.5 s. Calculate the momentum change.

Write down the values:

F = 6 N, Δt = 0.5 s, Δp = ?

Write down the equation:

FΔt = Δp

Substitute in the values, and calculate:

Δp = 6 × 0.5 = 3

The momentum changes by 3 kg·m/s: half a second of delivery, half of 6 kg·m/s.

A 12 N net force changes a scooter's momentum by 36 kg·m/s in 3 s. The force instead acts for only 1.5 s. Enter the momentum change in kg·m/s.

Your turn

A fixed net force acts on a cart. Which statement explains why the impulse FΔt equals the momentum change Δp?

A 7 N net force acts on a canoe for 2 s, changing its momentum by 14 kg·m/s. The same force acts for 6 s instead. Enter the new momentum change in kg·m/s.

Esther keeps the same 20 N net force on a cart but wants triple the momentum change. What should she change?