Momentum & Collisions
Momentum describes how much "oomph" a moving object has. It depends on both mass and velocity — a slow truck and a fast bullet can have similar momentum. Understanding momentum is key to predicting what happens when objects collide.
This guide covers linear momentum, the impulse-momentum theorem, conservation of momentum, elastic and inelastic collisions, worked examples, key formulas, and a 10-question practice quiz.
Momentum: p = mv
1What Is Momentum and Why Does It Matter?
Momentum is a measure of how hard it is to stop a moving object. It combines two things: how massive the object is and how fast it's going. A bowling ball rolling slowly and a tennis ball moving at incredible speed can have similar momentum.
Momentum is one of the most fundamental quantities in physics because it is conserved — in any collision or interaction where no external forces act, the total momentum before equals the total momentum after.
- Car crashes — crumple zones and airbags use momentum principles to save lives
- Billiard balls — momentum transfers when the cue ball hits other balls
- Rocket propulsion — exhaust gases go backward, rocket goes forward (conservation of momentum)
- Sports — a cricket bat or baseball bat follows through to maximize impulse
2Key Definitions
Momentum (p)
Product of mass and velocity: p = mv. A vector quantity (has direction). Units: kg·m/s
Impulse (J)
Change in momentum: J = FΔt = Δp. Also a vector. Units: N·s or kg·m/s
Elastic Collision
Both momentum and kinetic energy are conserved. Objects bounce off each other.
Inelastic Collision
Momentum is conserved but kinetic energy is NOT. Some KE becomes heat, sound, or deformation.
Perfectly Inelastic
Objects stick together after collision. Maximum KE loss. Momentum still conserved.
Isolated System
A system with no net external force. Total momentum of an isolated system is always conserved.
Kinetic Energy (KE)
Energy of motion: KE = ½mv2. A scalar (no direction). Not always conserved in collisions.
Coefficient of Restitution (e)
Ratio of relative speeds after/before collision. e = 1 (elastic), 0 < e < 1 (inelastic), e = 0 (perfectly inelastic).
3How Does Momentum Work?
Linear momentum is defined as the product of an object's mass and velocity. It tells you how much "motion" an object carries.
p = mv
p = momentum (kg·m/s)
m = mass (kg)
v = velocity (m/s)
Key Properties of Momentum
4Impulse and the Impulse–Momentum Theorem
Impulse is the change in momentum caused by a force acting over a time interval. Newton's second law can be rewritten in terms of momentum.
J = FΔt = Δp = mvf − mvi
J = impulse (N·s or kg·m/s)
F = average force (N)
Δt = time interval (s)
Δp = change in momentum (kg·m/s)
Why Does Impulse Matter?
The impulse-momentum theorem tells us that the same change in momentum can happen with a large force over a short time OR a small force over a long time. This is the principle behind many safety features:
Airbags
Increase Δt → decrease F
Crumple Zones
Car deforms slowly → lower peak force
Bending Knees
Landing with bent knees → longer stop time
5Conservation of Momentum
The law of conservation of momentum states that in an isolated system (no net external force), the total momentum before an interaction equals the total momentum after.
m1v1i + m2v2i = m1v1f + m2v2f
Total momentum before = Total momentum after
This law applies to all types of collisions — elastic, inelastic, and perfectly inelastic — as long as no external forces act on the system.
Interactive: Collision Simulator
Set masses and velocities, then watch momentum conservation in action.
Before Collision
After Collision
Kinetic Energy
Energy lost: 77.1%
Before
After
6What Are the Types of Collisions?
Elastic
Momentum: Conserved
Kinetic Energy: Conserved
Objects: Bounce apart
Examples: Billiard balls, atomic particles, ideal bouncy ball
e = 1
Inelastic
Momentum: Conserved
Kinetic Energy: NOT conserved
Objects: Bounce apart (deformed)
Examples: Car crash where cars separate, dropping a ball that doesn't bounce fully
0 < e < 1
Perfectly Inelastic
Momentum: Conserved
Kinetic Energy: Maximum loss
Objects: Stick together
Examples: Catching a ball, bullet embedding in a block, coupling train cars
e = 0
Momentum is always conserved in collisions (when no external forces act). Kinetic energy is only conserved in perfectly elastic collisions. Most real-world collisions are inelastic.
7Worked Examples
Example 1: Perfectly Inelastic Collision
A 2 kg cart moving at 3 m/s collides with and sticks to a 4 kg stationary cart.
Step 1: Momentum before: ptotal = m1v1 + m2v2 = 2(3) + 4(0) = 6 kg·m/s
Step 2: After collision, they stick together: combined mass = 2 + 4 = 6 kg
Step 3: By conservation: ptotal = 6 kg·m/s = 6 × vf
Step 4: vf = 6/6 = 1 m/s
Check KE loss: KEbefore = ½(2)(3²) = 9 J. KEafter = ½(6)(1²) = 3 J. Lost 6 J (67%)
Example 2: Elastic Collision (equal masses)
A 1 kg ball moving at 4 m/s collides elastically with a stationary 1 kg ball.
Step 1: For equal masses in elastic collision: first ball stops, second ball takes all the velocity
Step 2: v1f = ((m1−m2)/(m1+m2)) × v1 = ((1−1)/(1+1)) × 4 = 0 m/s
Step 3: v2f = (2m1/(m1+m2)) × v1 = (2/(2)) × 4 = 4 m/s
Check: Momentum: 1(4) = 1(0) + 1(4) = 4 kg·m/s ✓. KE: ½(1)(16) = ½(1)(16) = 8 J ✓
Example 3: Recoil Problem
A 60 kg skater throws a 5 kg ball at 10 m/s. How fast does the skater recoil?
Step 1: Initial momentum = 0 (both at rest)
Step 2: Conservation: 0 = mskatervskater + mballvball
Step 3: 0 = 60 × vskater + 5 × 10
Step 4: vskater = −50/60 = −0.83 m/s (backward)
Example 4: Impulse (Ball bouncing off a wall)
A 0.5 kg ball moving at 4 m/s hits a wall and bounces back at 3 m/s.
Step 1: Define right as positive. vi = +4 m/s, vf = −3 m/s
Step 2: Impulse = Δp = m(vf − vi) = 0.5(−3 − 4) = −3.5 N·s
Note: The negative sign means the impulse was directed to the left (away from the wall). The magnitude is 3.5 N·s.
8Key Formulas at a Glance
Momentum
p = mv
Impulse
J = FΔt
Impulse = Δp
J = Δp
Conservation
Σpbefore = Σpafter
Perfectly Inelastic
vf = (m1v1 + m2v2) / (m1 + m2)
Kinetic Energy
KE = ½mv2
| Concept | Formula |
|---|---|
| Momentum | p = mv |
| Impulse | J = FΔt = Δp |
| Newton's 2nd (momentum form) | F = Δp/Δt |
| Conservation of momentum | m1v1i + m2v2i = m1v1f + m2v2f |
| Perfectly inelastic | vf = (m1v1 + m2v2) / (m1 + m2) |
| Elastic (v₁ final) | v1f = ((m1−m2)v1 + 2m2v2) / (m1+m2) |
| Kinetic energy | KE = ½mv2 |
9Memory Aids
Think "p = mv" — the heavier or faster something is, the harder it is to stop. A parked truck has zero momentum; a moving mosquito has some!
In every collision — elastic, inelastic, perfectly inelastic — total momentum is conserved. Only kinetic energy is sometimes lost.
"Elastic" means everything bounces back — like a rubber band. Both momentum AND kinetic energy are conserved. "Inelastic" means something gets dented.
Same Δp, longer time = smaller force. That's why you bend your knees when landing, why airbags inflate, and why egg-drop contests use padding.
Momentum is a vector. Always define a positive direction first. Objects moving in opposite directions have opposite-sign momenta. Forgetting the sign is the #1 mistake.
10Common Mistakes Students Make
"Forgetting that momentum is a vector."
Direction matters! Always define a positive direction and use negative signs for objects moving the other way. A ball bouncing off a wall changes direction, so Δp is much larger than you might think.
"Assuming kinetic energy is always conserved."
Only elastic collisions conserve KE. In inelastic and perfectly inelastic collisions, KE is lost to heat, sound, and deformation. Momentum is always conserved, not energy.
"Not using the correct system boundary."
Momentum is only conserved for the entire system. If you only look at one object, it can gain or lose momentum from the other. Always include all interacting objects.
"Confusing momentum (p = mv) with kinetic energy (KE = ½mv²)."
Momentum is proportional to v (linear), while KE is proportional to v2 (quadratic). Doubling speed doubles momentum but quadruples kinetic energy.
"Not converting units before calculations."
Make sure mass is in kg, velocity in m/s, and time in seconds before plugging into formulas. A 10 g bullet is 0.01 kg!
Frequently Asked Questions
- Is momentum always conserved?
- Momentum is conserved in isolated systems where no net external force acts. In most collision problems, external forces are negligible during the brief collision, so momentum is effectively conserved.
- What is the difference between impulse and force?
- Force is instantaneous — it acts at a specific moment. Impulse is force multiplied by the time interval over which it acts (J = FΔt). A small force over a long time can produce the same impulse as a large force over a short time.
- Why do airbags reduce injury?
- Airbags increase the time of the collision, which reduces the peak force for the same change in momentum (impulse). Since J = FΔt = Δp, if Δt is larger, F must be smaller.
- What is the difference between elastic and inelastic collisions?
- In an elastic collision, both momentum and kinetic energy are conserved — objects bounce off each other. In an inelastic collision, momentum is conserved but kinetic energy is not — some energy is converted to heat, sound, or deformation. In a perfectly inelastic collision, the objects stick together.
- Can momentum be negative?
- Yes! Momentum is a vector quantity, so its sign depends on direction. If you define rightward as positive, an object moving leftward has negative momentum. This is crucial for solving collision problems correctly.
Practice Quiz
Test your understanding of momentum and collisions — select the correct answer for each question.
1.What is the momentum of a 5 kg object moving at 3 m/s?
2.What are the SI units of momentum?
3.A constant force of 20 N acts on an object for 4 seconds. What is the impulse delivered to the object?
4.A 4 kg cart moving at 6 m/s collides with a stationary 2 kg cart. If the 4 kg cart moves at 2 m/s after the collision, what is the velocity of the 2 kg cart?
5.In which type of collision is kinetic energy conserved?
6.A 3 kg object moving at 4 m/s collides with a 1 kg object at rest and they stick together. What is their combined velocity after the collision?
7.A 60 kg astronaut floating at rest in space throws a 5 kg tool at 12 m/s. What is the astronaut's recoil velocity?
8.Why do airbags and crumple zones reduce injuries in car crashes?
9.Two identical balls moving at the same speed collide head-on. Ball A moves east and Ball B moves west. What is the total momentum of the system?
10.A 0.5 kg ball moving at 8 m/s bounces off a wall and returns at 6 m/s in the opposite direction. What is the magnitude of the change in momentum?
Final Study Advice
- 1. Always define a positive direction before solving any momentum problem.
- 2. Draw a "before and after" diagram for every collision — label masses and velocities with signs.
- 3. Write the conservation equation first: total momentum before = total momentum after.
- 4. Identify the collision type (elastic, inelastic, perfectly inelastic) — this tells you whether KE is conserved too.
- 5. Check your answer: does the direction make sense? Is the speed reasonable?