Momentum and Impulse

What is Momentum?

🚛 Intuition of Momentum

①At the same speed, a truck is harder to stop than a bicycle — bigger mass

②At the same mass, faster is harder to stop

③Momentum p = mv — the "amount of motion" combining mass and velocity

Collision and Momentum Conservation

Ball 1 mass m₁5 kg

Ball 1 velocity v₁6 m/s

💡 Conservation of Momentum

①Without external force, total momentum is unchanged through collision

②m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂' (conserved!)

③Larger mass → smaller change in velocity after collision

Impulse and Change in Momentum

Momentum

p = mv

Momentum (kg·m/s) = mass (kg) × velocity (m/s)

Impulse = Change in Momentum

I = FΔt = Δp = mv' - mv

Force × time = change in momentum

🥊 Meaning of Impulse

①Same Δp can be made by: large F × short Δt = small F × long Δt

②Airbag idea: extend collision time to reduce force!

③Why a baseball catcher pulls the glove back: Δt ↑ → F ↓

Momentum Conservation Formula

Conservation of Momentum

m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'

Total momentum is preserved when external force = 0

Perfectly Inelastic (Stick Together)

v' = m₁v₁ + m₂v₂m₁ + m₂

Two objects merge into one

Worked Examples

Example 1

A 0.5 kg ball hits a wall at 4 m/s and bounces back at 2 m/s in the opposite direction. What is the magnitude of the impulse on the ball?

Impulse = change in momentum I = mv′ − mv. Assign signs by direction (incoming +, rebound −).

I = mv' − mv

Substitute v = +4, v′ = −2.

0.5 × (−2) − 0.5 × 4 = −3 kg·m/s

▸ magnitude 3 kg·m/s (opposite to the initial motion)

Momentum and impulse are vectors. A rebound reverses direction, so the sign must be included.

Example 2

A 0.15 kg baseball arrives at 40 m/s, is hit, and leaves at 60 m/s in the opposite direction. If contact lasts 0.01 s, what is the average force?

Momentum change Δp = m(v′ − v). Opposite directions, so v = −40, v′ = +60.

Δp = 0.15 × (60 − (−40)) = 15 kg·m/s

Since impulse = FΔt = Δp, F = Δp/Δt.

F = ΔpΔt = 150.01 = 1500 N

▸ 1500 N

For the same Δp, increasing the contact time Δt reduces the force (the principle of airbags and giving with the glove).

Summary

Momentum

p = mv

Impulse

I = FΔt = Δp

CSAT-style

A 2 kg object moving at 4 m/s collides with a stationary 2 kg object and they stick together. What is the speed after the collision?

①1 m/s

②2 m/s

③3 m/s

④4 m/s

⑤6 m/s

▸ ② 2 m/s

For a perfectly inelastic collision (sticking), use momentum conservation to find the common speed.

v' = m₁v₁ + m₂v₂m₁ + m₂

Substitute m₁=2, v₁=4, m₂=2, v₂=0.

v' = 2×4 + 2×02+2 = 84 = 2 m/s

🎯 Exam Points

①Momentum p = mv is a vector — has direction!

②Impulse I = FΔt = Δp (change in momentum)

③External force = 0 → momentum is conserved (collisions, explosions)

④Airbag/cushion: deliver same Δp with larger Δt → smaller F

⑤Perfectly inelastic: v' = (m₁v₁+m₂v₂)/(m₁+m₂) after merging

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