Unit 07 - Grade 11-12 Physics

Momentum and Collisions

Learn how momentum explains impacts, collisions, explosions, recoil, safety systems, and motion when objects interact over short time intervals.

Lesson roadmap

What Students Should Master in This Unit

Momentum is a powerful conservation quantity. It helps students analyze collisions and explosions even when forces are large, complicated, and act for only a short time.

Calculate momentum and impulse

Use mass, velocity, force, and time to describe motion changes during interactions.

Apply conservation laws

Use total system momentum before and after a collision or explosion.

Classify collisions

Compare elastic, inelastic, and perfectly inelastic collisions using momentum and energy.

Quick navigation

Jump to a Topic

Momentum Impulse Impulse Graphs Conservation of Momentum Collision Types Elastic Collisions Inelastic Collisions Explosions and Recoil Center of Mass PhET Collision Lab Worked Examples Practice Problems
Motion quantity

1. Momentum

Momentum measures how much motion an object has. It depends on mass and velocity, so a large slow object and a small fast object can both have significant momentum.

Momentum depends on mass and velocity Momentum depends on both mass and velocity small m large m v v p = mv
A heavier object and a faster object both carry more momentum, so signs and direction matter in collision problems.
Momentum p = mv Momentum is a vector because velocity is a vector.
Momentum unit kg·m/s Equivalent to N·s.
Change in momentum Δp = pf - pi Final momentum minus initial momentum.

Momentum Direction

In one-dimensional problems, choose one direction as positive. Momentum moving the opposite way is negative. Direction signs are essential in collision problems.

Common mistake: Momentum is not the same as kinetic energy. Momentum depends on v, while kinetic energy depends on v2.
Force over time

2. Impulse

Impulse describes how a force acting over time changes momentum. The same momentum change can happen with a large force for a short time or a smaller force for a longer time.

Impulse changes momentum Impulse changes momentum during contact time pi J = Δp contact force FavgΔt = m(vf - vi)
Increasing collision time reduces average force for the same change in momentum.
Impulse J = FavgΔt For average force over a time interval.
Impulse-momentum theorem J = Δp Impulse equals change in momentum.
Expanded form FavgΔt = m(vf - vi) Useful for safety and impact problems.

Safety Connection

Airbags, helmets, pads, and crumple zones increase collision time. For the same momentum change, increasing time reduces average force.

Favg = Δp / Δt
Area under force graph

3. Impulse from Force-Time Graphs

Impulse is the area under a force-time graph. This is useful when force changes during a collision.

Impulse from graph area Area under an F vs. t graph gives impulse F t J = area triangle area = (1/2)base height
Force-time graph area has units of newton-seconds, which equals momentum change.
Graph impulse J = area under F vs. t graph Area units are N·s.
Rectangle area J = FΔt Constant force.
Triangle area J = (1/2)base·height Linearly changing force.
Connection: Force-displacement graph area gives work. Force-time graph area gives impulse.
Closed systems

4. Conservation of Momentum

Total momentum of a system is conserved when the net external impulse on the system is zero or negligible. During a short collision, internal forces between objects can be large, but they cancel within the system.

Conservation of momentum In an isolated system, total momentum before equals total momentum after before after ptotal,i = ptotal,f
Internal forces during the collision cancel as a pair, so the system momentum stays constant when external impulse is negligible.
Conservation statement ptotal,i = ptotal,f Total momentum before equals total momentum after.
Two-object form m1v1i + m2v2i = m1v1f + m2v2f Use signs for direction.
System condition ΣJexternal = 0 Momentum is conserved when external impulse is zero.

When Momentum Is Conserved

  • Objects collide on a nearly frictionless track.
  • An explosion occurs with negligible external force during the event.
  • Two skaters push apart on ice.
  • A cart system is analyzed over a short collision time.
Classifying interactions

5. Types of Collisions

Momentum is conserved in all isolated collisions, but kinetic energy may or may not be conserved.

Collision type comparison Collision type depends on what happens after impact Elastic bounce; K conserved Inelastic deform; K decreases Perfectly inelastic stick together after impact
Momentum is conserved in all isolated collisions, but kinetic energy is conserved only in elastic collisions.
Collision Type Momentum Kinetic Energy Object Behavior
Elastic Conserved Conserved Objects bounce apart with no kinetic energy loss in the ideal model.
Inelastic Conserved Not conserved Objects deform, heat, or sound may be produced.
Perfectly inelastic Conserved Not conserved Objects stick together and move with one final velocity.
Important: Kinetic energy can decrease in a collision, but total energy is still conserved when thermal, sound, and deformation energy are included.
Bounce apart

6. Elastic Collisions

In an elastic collision, both momentum and kinetic energy are conserved. These are idealized collisions, but they are useful for carts, low-friction systems, and molecular-scale interactions.

Elastic collision velocity exchange Elastic collisions conserve momentum and kinetic energy before after Ki = Kf equal masses often exchange velocities
For equal masses where one starts at rest, an elastic collision often looks like velocity transfer from one object to the other.
Momentum conserved pi = pf Use signs for velocity.
Kinetic energy conserved Ki = Kf Only for elastic collisions.
Equal masses, one at rest objects often exchange velocities A useful special case.
Stick or deform

7. Inelastic and Perfectly Inelastic Collisions

In inelastic collisions, momentum is conserved but kinetic energy is not. In a perfectly inelastic collision, objects stick together after impact.

Perfectly inelastic collision Perfectly inelastic collisions stick together before after shared vf momentum conserved, kinetic energy decreases
After a perfectly inelastic collision, both objects move together with one final velocity.
Perfectly inelastic form m1v1i + m2v2i = (m1 + m2)vf Objects share one final velocity.
Final shared velocity vf = [m1v1i + m2v2i] / (m1 + m2) Use signs for direction.
Kinetic energy lost ΔK = Kf - Ki Usually negative in inelastic collisions.
Objects push apart

8. Explosions and Recoil

In an explosion, objects start together and push apart. Momentum is conserved if external impulse is negligible, even though kinetic energy often increases because stored energy is converted into motion.

Explosion and recoil momentum Objects starting together push apart with equal opposite momentum p1 p2 ptotal,i = 0, so ptotal,f = 0
Recoil problems use the same conservation idea: momenta balance in opposite directions.
Starting from rest ptotal,i = 0 Total final momentum must also be zero.
Two-piece explosion m1v1f + m2v2f = 0 Pieces move in opposite directions.
Recoil relationship mobjectvobject = -mrecoilvrecoil Equal magnitude, opposite direction momenta.

Examples

  • A cannon recoils when a cannonball fires forward.
  • A person jumps from a small boat and the boat moves backward.
  • Two skaters push apart from rest.
  • A spring-loaded cart pushes two carts apart.
System motion

9. Center of Mass Thinking

The center of mass is the average position of a system's mass. In the absence of external net force, the center of mass moves with constant velocity even if the objects inside the system collide or explode.

Center of mass Center of mass is the balance point of a system m1 m2 xcm center of mass shifts closer to the larger mass
Internal collisions can be messy, but the center of mass motion is simple when external force is negligible.
Two-object center of mass xcm = (m1x1 + m2x2) / (m1 + m2) Position weighted by mass.
Center of mass velocity vcm = ptotal / mtotal System momentum divided by total mass.
No external force vcm = constant Internal collisions do not change total system motion.
Simulation lab

10. PhET Collision Lab

The official PhET Collision Lab lets students test one-dimensional and two-dimensional collisions, compare momentum before and after, and investigate how kinetic energy changes in different collision types.

Collision simulation workflow Use Collision Lab to compare before-and-after momentum set variables m, v, elasticity run collision record data p before and after compare total momentum
Students should change one variable at a time, record before-and-after values, and decide whether momentum and kinetic energy were conserved.
Collision Lab

Use carts or balls to investigate conservation of momentum, elastic collisions, inelastic collisions, mass ratios, velocity signs, and kinetic energy changes.

Lab idea: run three trials with different masses, then compare total momentum before and after each collision.
Open PhET Lab
Suggested Investigation

Set one object at rest, vary the moving object's mass and speed, then record initial momentum, final momentum, and kinetic energy before and after collision.

Question: Which quantity is always conserved in an isolated collision?
Use Practice Problems
Investigation skills

11. Momentum and Collision Lab Skills

Momentum labs usually involve carts, tracks, motion sensors, video analysis, or collision simulations. Students should compare total momentum before and after an interaction.

Momentum lab setup A good collision lab measures mass and velocity before and after sensor sensor data table measure m, vi, vf, and collision time
Professional analysis compares total initial and final momentum, then explains percent difference and sources of error.

Common Lab Measurements

  • Mass of each object in kilograms.
  • Initial and final velocity of each object.
  • Collision time for impulse problems.
  • Force-time data if a force sensor is available.
  • Kinetic energy before and after collision.

Lab Analysis Questions

  • Was the system isolated enough for momentum to be conserved?
  • How close were initial and final total momentum values?
  • Was kinetic energy conserved, lost, or gained?
  • What sources of error affected the result?
  • Did friction or track alignment affect the collision?
Worked examples

12. Worked Examples

Example 1: Momentum

A 0.150 kg baseball moves at 40.0 m/s. Find its momentum.

p = mv = (0.150)(40.0) = 6.00 kg·m/s.

Example 2: Impulse

A 1200 kg car changes velocity from 20.0 m/s to 5.0 m/s. Find impulse on the car.

J = Δp = m(vf - vi).

J = 1200(5.0 - 20.0) = -18,000 N·s.

The negative sign means impulse is opposite the initial direction of motion.

Example 3: Perfectly inelastic collision

A 2.0 kg cart moving at 6.0 m/s collides with a 3.0 kg cart at rest. They stick together. Find final velocity.

m1v1i + m2v2i = (m1 + m2)vf.

(2.0)(6.0) + (3.0)(0) = (5.0)vf.

vf = 2.4 m/s.

Example 4: Explosion from rest

Two skaters push apart from rest. A 50 kg skater moves left at 2.0 m/s. A 75 kg skater moves right. Find the rightward speed.

Initial momentum is zero. Choose right as positive.

(50)(-2.0) + (75)v = 0.

v = 1.33 m/s right.

Example 5: Average force from impulse

A 0.060 kg tennis ball changes velocity from -30 m/s to +25 m/s during contact lasting 0.010 s. Find average force.

Δp = m(vf - vi) = 0.060[25 - (-30)] = 3.3 N·s.

Favg = Δp / Δt = 3.3/0.010 = 330 N.

Example 6: Kinetic energy change

A 4.0 kg cart moving at 3.0 m/s sticks to a 4.0 kg cart at rest. Find final speed and kinetic energy lost.

Momentum: (4.0)(3.0) = (8.0)vf, so vf = 1.5 m/s.

Ki = (1/2)(4.0)(3.0)2 = 18 J.

Kf = (1/2)(8.0)(1.5)2 = 9 J.

Kinetic energy lost = 9 J.

Independent practice

13. Practice Problems

Try each problem first. Use signs carefully and clearly identify the system before checking the answer.

1. Find the momentum of a 5.0 kg object moving at 4.0 m/s east.

Answer

p = mv = (5.0)(4.0) = 20 kg·m/s east.

2. A 0.20 kg ball moving at 15 m/s has what momentum?

Answer

p = (0.20)(15) = 3.0 kg·m/s.

3. A 2.0 kg object changes velocity from 3.0 m/s to 9.0 m/s. Find change in momentum.

Answer

Δp = m(vf - vi) = 2.0(9.0 - 3.0) = 12 kg·m/s.

4. A 10 N force acts for 0.50 s. Find impulse.

Answer

J = FΔt = (10)(0.50) = 5.0 N·s.

5. A 1000 kg car receives an impulse of -5000 N·s. Find its change in velocity.

Answer

J = mΔv, so Δv = J/m = -5000/1000 = -5.0 m/s.

6. A force-time graph is a triangle with base 0.20 s and height 80 N. Find impulse.

Answer

J = (1/2)(0.20)(80) = 8.0 N·s.

7. A 3.0 kg cart moving at 2.0 m/s sticks to a 1.0 kg cart at rest. Find final velocity.

Answer

(3.0)(2.0) + (1.0)(0) = (4.0)vf, so vf = 1.5 m/s.

8. A 4.0 kg cart moving right at 3.0 m/s hits a 2.0 kg cart moving left at 1.0 m/s. They stick. Find final velocity.

Answer

Choose right positive. Initial momentum = (4.0)(3.0) + (2.0)(-1.0) = 10 kg·m/s.

Total mass = 6.0 kg. vf = 10/6.0 = 1.67 m/s right.

9. Two skaters push apart from rest. One has mass 40 kg and moves right at 3.0 m/s. The other has mass 60 kg. Find the second skater's velocity.

Answer

Initial momentum is zero. (40)(3.0) + (60)v = 0.

v = -2.0 m/s, so 2.0 m/s left.

10. A rifle of mass 4.0 kg fires a 0.020 kg bullet at 600 m/s. Find recoil speed of the rifle.

Answer

Initial momentum is zero. (0.020)(600) + (4.0)v = 0.

v = -3.0 m/s. The rifle recoils at 3.0 m/s backward.

11. In an isolated collision, total momentum before is 12 kg·m/s. What is total momentum after?

Answer

12 kg·m/s.

12. In a perfectly inelastic collision, two objects do what after impact?

Answer

They stick together and move with the same final velocity.

13. Which collision type conserves both momentum and kinetic energy?

Answer

Elastic collision.

14. A 2.0 kg object at x = 0 m and a 6.0 kg object at x = 4.0 m. Find center of mass.

Answer

xcm = [(2.0)(0) + (6.0)(4.0)]/(8.0) = 3.0 m.

15. Why do airbags reduce injury force?

Answer

They increase stopping time, so the same momentum change happens with a smaller average force.

16. In PhET Collision Lab, what should stay the same before and after an isolated collision?

Answer

Total momentum of the system.

Final review

14. What to Know Before Moving On

  • Momentum is p = mv and is a vector.
  • Impulse is J = FavgΔt and equals change in momentum.
  • Area under a force-time graph gives impulse.
  • Momentum is conserved when net external impulse is zero or negligible.
  • Elastic collisions conserve momentum and kinetic energy.
  • Inelastic collisions conserve momentum but not kinetic energy.
  • Perfectly inelastic collisions stick together after impact.
  • Explosions and recoil conserve total momentum when external impulse is negligible.
  • Always choose a positive direction and keep velocity signs.
Previous Unit: Work, Energy, and Power Back to Unit Library Next Unit: Circular Motion and Gravitation

Useful Links

Physics Units

Support

© 2025 Physicstutors.org | All rights reserved