Momentum and Impulse

Momentum is mass times velocity, which is why a truck at the same speed is harder to stop than a bicycle. With no external force, total momentum is conserved in a collision, and impulse equals force times time, the change in momentum. An airbag lengthens collision time so the force drops, the same impulse idea. Change the two balls' masses and speeds here and watch how momentum is shared before and after the hit.

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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Newton's Laws

Conservation of Mechanical Energy

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