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Grade 10-11 (age 15-17)

high school physics Momentum and Impulse

Momentum is mass times velocity, p = mv, so a heavy truck is far harder to stop than a bicycle at the same speed. When no outside force acts, the total momentum before and after a collision stays the same, and the impulse, force multiplied by the time it acts, equals the change in momentum. An airbag lowers the force on the body by stretching out the time of the impact, which follows from the same impulse relation. In the picture, a moving ball hits a 3 kg ball at rest and the two stick together; whatever mass and speed you choose, the total momentum comes out the same before and after.

What Momentum Is

🚛 How Hard It Is to Stop
①At the same speed, a truck with more mass is harder to stop than a bicycle
②At the same mass, the faster object is harder to stop
③This amount of motion, combining mass and velocity, is momentum p = mv, a vector with a direction just like velocity

Collisions and Conservation of Momentum

5 kg
6 m/s
💡 Reading the Picture
①Before the collision, the only momentum is ball 1's p₁ = m₁v₁; ball 2 at rest has p₂ = 0
②After the collision, the momentum of the combined body, p', equals that total
③Because the mass grows to m₁ + m₂, the speed drops to v' = p/(m₁ + m₂), so the purple arrow is shorter than the red one
④When m₁ is much larger than ball 2's 3 kg, v' comes close to v₁; when it is small, v' drops a lot

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
The force on an object multiplied by the time it acts equals the change in momentum
🥊 Same Impulse, Different Force
①The same change in momentum can come from a large force for a short time or a small force for a long time
②An airbag stretches out the impact time, lowering the average force on the body
③A catcher pulls the glove back while catching to make the contact time longer and the force on the hand smaller

Conservation Formulas

Conservation of Momentum
m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'
With zero external force, the total momentum of the two objects is the same before and after the collision
Perfectly Inelastic Collision (Stick Together)
v' = m₁v₁ + m₂v₂m₁ + m₂
The two objects stick and move with a common velocity v', which is the "after" side of the picture

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?
1
Impulse is the change in momentum, I = mv' − mv. Take the original direction of motion as + and the rebound direction as −.
I = mv' − mv
2
Substitute v = +4 and v' = −2.
0.5 × (−2) − 0.5 × 4 = −3 kg·m/s
▸ magnitude 3 kg·m/s (opposite to the initial motion)
Ignoring the signs and computing 0.5 × 2 − 0.5 × 4 gives a magnitude of 1, which is wrong. In a rebound, the reversal of direction makes the change in momentum larger.
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?
1
Taking the incoming direction as −, v = −40 and v' = +60, so Δp = m(v' − v).
Δp = 0.15 × (60 − (−40)) = 15 kg·m/s
2
Since the impulse FΔt equals Δp, F = Δp/Δt.
F = ΔpΔt = 150.01 = 1500 N
▸ 1500 N
The ball stays on the bat for only 0.01 s, so the average force is about 1000 times the ball's weight.

Summary

Momentum
p = mv
Impulse
I = FΔt = Δp
CSAT-style problem, adapted
Moving at 4 m/s, a 2 kg object collides with a 2 kg object at rest, and the two stick together. What is their speed after the collision?
①1 m/s
②2 m/s
③3 m/s
④4 m/s
⑤6 m/s
▸ ② 2 m/s
1
The two objects become one body in a perfectly inelastic collision, so use momentum conservation to find the common speed.
v' = m₁v₁ + m₂v₂m₁ + m₂
2
Substitute m₁ = 2, v₁ = 4, m₂ = 2 and v₂ = 0.
v' = 2×4 + 2×02+2 = 84 = 2 m/s
🎯 Exam Points
①Momentum p = mv is a vector, so motion in the opposite direction gets a negative sign
②Impulse I = FΔt equals the change in momentum Δp
③With zero external force, total momentum is conserved in collisions and break-ups
④Airbags and cushions spread the same Δp over a longer Δt to lower the average force F
⑤In a perfectly inelastic collision, the common velocity is v' = (m₁v₁ + m₂v₂)/(m₁ + m₂)
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Newton's Laws of Motion
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Conservation of Mechanical Energy

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