AP Physics 1 momentum and collisions
Momentum explains collisions, explosions and anything where forces act for a short time. Here is Unit 4 of AP Physics 1.
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Momentum and impulse
- Momentum p = mv is a vector.
- Impulse J = FΔt equals the change in momentum.
- The area under a force-time graph is the impulse.
Airbags and crumple zones work by increasing the collision time, which reduces the force for the same change in momentum.
Conservation of momentum
If no net external force acts on a system, its total momentum stays the same. Define the system first; it usually includes every object in the collision.
Types of collisions
- Elastic: momentum and kinetic energy are both conserved.
- Inelastic: momentum is conserved, but some kinetic energy is converted to other forms.
- Perfectly inelastic: the objects stick together and move with a common velocity.
Centre of mass
With no net external force, the centre of mass of a system moves at constant velocity, even during collisions and explosions.
Problem-solving steps
- Choose a positive direction.
- Write total momentum before and after.
- Set them equal and solve.
- Use energy only if the question says the collision is elastic.
Common mistakes
- Forgetting signs for objects moving in opposite directions.
- Assuming kinetic energy is conserved in every collision.
Worked examples
Example 1: an elastic collision check
A 1 kg ball at 4 m/s hits an identical ball at rest. After the collision, the first ball stops and the second moves at 4 m/s. Is the collision elastic?
- Momentum: 4 before and 4 after. Kinetic energy: 8 J before and 8 J after. Yes, it is elastic.
Example 2: a recoil
A 60 kg skater throws a 2 kg ball forward at 15 m/s while standing still. Find the skater's recoil speed.
- 0 = 2 × 15 + 60v, so v = −0.5 m/s: 0.5 m/s backward.
Example 3: impulse from a graph
A force-time graph shows a triangle peaking at 200 N over 0.04 s. Find the change in momentum.
- Impulse = area = ½ × 0.04 × 200 = 4 N·s, so Δp = 4 kg·m/s.
Example 4: a bouncing ball
A 0.2 kg ball hits a wall at 10 m/s and bounces straight back at 8 m/s. Find the impulse.
- Taking the original direction as positive: Δp = 0.2(−8) − 0.2(10) = −3.6 kg·m/s. A bounce gives a bigger impulse than stopping, because the direction reverses.
Practice questions
Try each one before opening the answer.
Q1. A 3 kg cart at 2 m/s hits a 1 kg cart moving toward it at 2 m/s. They stick. Find the final velocity.
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Q2. Why do airbags reduce injury?
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Q3. Two objects collide. Momentum is conserved but kinetic energy decreases. What type of collision is it?
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Frequently asked questions
Is momentum always conserved?
Only for a system with no net external force. Choose a system that includes both colliding objects; during a short collision, external forces like friction are usually negligible.
Can kinetic energy increase in a collision?
Yes, in explosions: stored chemical or spring energy is converted to kinetic energy, while total momentum still stays the same.
The key ideas in more depth
Why momentum is so useful
During a collision, the forces between objects are large, brief and hard to measure. Momentum lets you skip them. Because the internal forces come in Newton's third law pairs, they cancel when you add up the whole system. Only external forces can change the system's total momentum, and during a short collision they are usually small. That is why "momentum before equals momentum after" works so well.
Two dimensions
Momentum is a vector, so in two-dimensional collisions it is conserved separately in each direction. Split each velocity into x and y components, write one conservation equation for each direction, and solve them together.
The centre of mass
If no net external force acts, the centre of mass of a system keeps moving at constant velocity, even if the objects collide or explode. A firework that explodes in mid-air sends pieces in every direction, but the centre of mass keeps following the original path until pieces hit the ground.
Exam tips
- Always state the system and why its momentum is conserved.
- Write the conservation equation with symbols before numbers.
- For "is it elastic?" questions, calculate kinetic energy before and after.
- For force-time graphs, the area is the impulse; for momentum-time graphs, the slope is the force.
More practice
Try each one before opening the answer.
Q1. A 2 kg ball moving east at 3 m/s collides with a 1 kg ball moving north at 4 m/s. They stick together. Find the final velocity components.
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Q2. A momentum-time graph for an object is a straight line rising from 2 to 8 kg·m/s over 3 s. Find the net force.
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One more worked example
Example 5: a ballistic pendulum
A 0.01 kg bullet hits a 1.99 kg block hanging at rest and sticks. The block rises 0.2 m. Find the bullet's speed.
- After the collision, energy is conserved as the block swings up: v = √(2gh) = √(2 × 9.8 × 0.2) ≈ 1.98 m/s.
- During the collision, momentum is conserved: 0.01u = 2.00 × 1.98, so u ≈ 396 m/s.
- Momentum for the collision, energy for the swing: never the other way round.
Extra practice
Try each one before opening the answer.
Q1. A 1000 kg car at 20 m/s brakes to a stop in 4 s. Find the average braking force.
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Q2. Can a system have zero momentum but non-zero kinetic energy?
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Key takeaways
- Momentum p = mv is a vector; directions and signs matter.
- Impulse equals the change in momentum and the area under a force-time graph.
- Total momentum is conserved when the net external force on the system is zero.
- Kinetic energy is conserved only in elastic collisions.
- Use momentum for the collision itself and energy for motion before or after it.
Quick check
A 4 kg object at rest explodes into two pieces. A 1 kg piece flies left at 6 m/s. Find the velocity of the 3 kg piece.