Momentum & Impulse

You’ll be able to

Compute momentum as p = mv and treat it as a vector
Apply the impulse–momentum theorem, J = FΔt = Δp
Explain how extending the contact time reduces the force

Momentum: mass in motion

Momentum is the product of an object’s mass and velocity, p = mv, measured in kg·m/s. It is a vector, pointing the same way as the velocity, so its sign follows your chosen positive direction. A heavy truck rolling slowly and a light bullet moving fast can carry the same momentum. Momentum captures "how hard it is to stop" something in a way that speed alone cannot.

Impulse changes momentum

To change an object’s momentum you apply a force over a time interval. The impulse J = FΔt equals the change in momentum, Δp. Rearranged, this is why a large force over a short time and a small force over a long time can produce the same change. Airbags, crumple zones, and bending your knees on landing all work by stretching Δt, which lowers the force F needed for the same Δp.

Momentum

p = mv

A vector in kg·m/s. Its direction is the direction of motion; reverse the motion and the sign flips.

Impulse–momentum theorem

J = F·Δt = Δp = m·v_f − m·v_i

Impulse (N·s) equals the change in momentum. For a bounce, remember v_f and v_i point in opposite directions.

Worked example

A 2 kg ball moving at 5 m/s is struck and continues in the same direction at 9 m/s. If the bat is in contact for 0.1 s, find the impulse and the average force.

1.Momentum change: Δp = m(v_f − v_i) = 2 × (9 − 5) = 2 × 4 = 8 kg·m/s.
2.Impulse equals that change: J = Δp = 8 N·s.
3.Average force: F = J ÷ Δt = 8 ÷ 0.1.
4.Compute: F = 80 N in the direction of motion.

Answer: J = 8 N·s and F = 80 N

Tip

For a ball that bounces back, the velocity reverses sign. A ball hitting a wall at +8 m/s and returning at −8 m/s has Δv = −16 m/s, not zero — the reversal is what makes the impulse large.

Checkpoint

A 3 kg ball moves at 4 m/s. What is the magnitude of its momentum?

Checkpoint

A 0.5 kg ball hits a wall at 8 m/s and bounces straight back at 8 m/s. What is the magnitude of the impulse the wall delivers to the ball?

On the exam

When asked why a safety feature reduces force, argue from J = FΔt with Δp fixed. The impulse (momentum change) is set by the collision, so a longer Δt forces a smaller F. Name the theorem to earn the reasoning point.

Answer the 2 checkpoints as you read.