Collisions and Momentum

Theory — Momentum and Its Conservation

Linear Momentum

Momentum is the product of an object's mass and its velocity. It is a vector — its sign carries the direction of motion. A heavy, fast object has a large momentum and is hard to stop; a light or slow one has little. Momentum, not speed, measures "how much motion" an object carries.

Linear Momentum p = m · v
Units: kilogram-metres per second (kg·m/s)
Sign convention: rightward positive, leftward negative

Conservation of Momentum

When two objects interact and no outside (external) force acts on the pair, the total momentum of the system is unchanged by the interaction. During a collision the two objects push on each other with equal and opposite forces (Newton's third law), so whatever momentum one gains, the other loses. The total before equals the total after.

Conservation of Momentum (two bodies) m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
(u = velocity before, v = velocity after)

Total momentum is conserved in every collision

Elastic vs Inelastic Collisions

Collisions differ in what happens to the kinetic energy. In an elastic collision the objects bounce apart and the total kinetic energy is conserved. In an inelastic collision some kinetic energy is converted into heat, sound, or deformation, so the total kinetic energy decreases. In a perfectly inelastic collision the objects stick together and move as one — the maximum kinetic energy is lost (though momentum is still conserved).

Kinetic Energy KE = ½ m v² (always ≥ 0; not a vector)

Elastic: ΣKE_before = ΣKE_after
Inelastic: ΣKE_after < ΣKE_before (energy lost to heat/sound)
Perfectly inelastic: objects stick → v = (m₁u₁ + m₂u₂)/(m₁+m₂)

Momentum: always conserved · Kinetic energy: only in elastic

Elastic collision

Objects bounce apart. Momentum conserved AND kinetic energy conserved. Idealized — like hard spheres or air-track gliders with bumpers.

Inelastic collision

Objects may stick or deform. Momentum conserved, but kinetic energy decreases (lost to heat, sound, deformation). Perfectly inelastic = stick together.

Quantity	Elastic	Inelastic	Perfectly inelastic
Total momentum	conserved	conserved	conserved
Total kinetic energy	conserved	decreases	decreases (maximum loss)
After collision	bounce apart	bounce / partly stick	move together as one

Instructions — Running the Virtual Experiment

The Collision Lab tab lets you set up and watch collisions; the Data Table tab logs the before-and-after momentum and energy. Record every reading in your lab notebook.

Experiment 1 — An Elastic Collision (Collision Lab tab)

Open Simulation → Collision Lab. Choose collision type Elastic. Set the two masses and velocities with the sliders (each velocity can be positive or negative, so the balls can move in either direction) — or click the Load scenario button to load the 2 kg @ −1.5 m/s and 3 kg @ +3 m/s setup directly. The readout shows velocity, momentum, and kinetic energy for each ball separately and as totals, before and after.

Press Launch. Read the total momentum and total kinetic energy in the "before" and "after" panels. Confirm both are equal before and after — an elastic collision conserves both.

Experiment 2 — A Perfectly Inelastic Collision (Collision Lab tab)

Switch the collision type to Inelastic (stick) with the same masses and speeds. Launch again.

The balls now move off together. Confirm that the total momentum is still conserved, but the total kinetic energy is less than before. Record how much kinetic energy was lost and note that this energy became heat, sound, and deformation.

Experiment 3 — Conservation Across Many Trials (Data Table tab)

Open Data Table. Choose a collision type, set the masses and initial velocities, and click Run & record. The row logs p_before, p_after, KE_before, and KE_after.

Run several trials, mixing elastic and inelastic types and different masses. Click Check conservation. Confirm p_before = p_after for every row, while KE is conserved only for the elastic rows.

Simulation — Two-Ball Collisions

ball 1 (pink)

ball 2 (blue)

Ball size scales with mass · arrow length with velocity.

Setup

Collision type

Ball 1 (pink) mass 2 kg

Ball 1 (pink) velocity −1.5 m/s

Ball 2 (blue) mass 3 kg

Ball 2 (blue) velocity +3 m/s

Before / After

v₁ (pink) now— m/s

v₂ (blue) now— m/s

p₁ (pink)—

p₂ (blue)—

total p— kg·m/s

KE₁ (pink)—

KE₂ (blue)—

total KE— J

Press Launch.

The bars compare total momentum & KE before vs after for the latest run.

Trial setup

Collision type

m₁ 2 kg

u₁ +3 m/s

m₂ 1 kg

u₂ 0 m/s

Type	p before	p after	KE before	KE after
No trials yet — set up a collision and click "Run & record".

Team Questions

Question 1. A 3 kg ball moves to the right at 4 m/s. What is its momentum? (Type just the number in kg·m/s)

Question 2. Which quantity is conserved in every collision, whether elastic or inelastic — momentum or kinetic energy? (Answer "momentum" or "kinetic energy")

Question 3. A 4 kg ball at 5 m/s strikes a 6 kg ball at rest and they stick together (perfectly inelastic). What is their common velocity afterward? Use v = (m₁u₁ + m₂u₂)/(m₁ + m₂). (Type just the number in m/s)

Question 4. In that perfectly inelastic collision, is the total kinetic energy after the collision equal to, less than, or greater than before? (Answer "less", "equal", or "greater")

Question 5. A moving billiard ball strikes an identical stationary ball head-on in an elastic collision. Describe what happens: the moving ball ___ and the struck ball ___ . (In one or two words — what does the first ball do?)

Question 6. A 2 kg ball at +3 m/s collides elastically with a 1 kg ball at rest. Total momentum before is 6 kg·m/s. What must the total momentum be after? (Type just the number in kg·m/s)

Question 7 — Challenge. A 1500 kg car traveling east at 20 m/s collides with and locks onto a 2500 kg truck at rest. What is the velocity of the wreckage immediately after? (Type just the number in m/s)

Example Lab Report

Sample report demonstrating the expected format and level of detail. Use as a guide for your own submission.

Collisions and Momentum: Conservation of Momentum and Kinetic Energy

Physics | Section: [Your Section] | Date: [Date]

Lab Members: [Names of all members present]

Purpose

To investigate one-dimensional collisions between two balls and to verify that total linear momentum is conserved in every collision, while total kinetic energy is conserved only in elastic collisions. The lab compares elastic and perfectly inelastic collisions for a range of masses and velocities.

Theory

When no external force acts on a two-body system, the total momentum before a collision equals the total momentum after. Kinetic energy, however, is conserved only when the collision is elastic; in inelastic collisions some kinetic energy is converted to heat, sound, and deformation.

p = mv · KE = ½mv²
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂ (always)
Perfectly inelastic: v = (m₁u₁ + m₂u₂)/(m₁ + m₂)

For the elastic case the two final velocities are found from conservation of both momentum and kinetic energy; for the perfectly inelastic case the two balls share one final velocity.

Calculations — Sample: m₁ = 2 kg at u₁ = +3 m/s, m₂ = 1 kg at rest

Momentum before: p = (2)(3) + (1)(0) = 6 kg·m/s

KE before: KE = ½(2)(3²) + 0 = 9 J

Elastic after: v₁ = [(m₁−m₂)u₁]/(m₁+m₂) = (1·3)/3 = 1 m/s; v₂ = (2m₁u₁)/(m₁+m₂) = (12)/3 = 4 m/s. Check: p = (2)(1)+(1)(4) = 6 kg·m/s ✓; KE = ½(2)(1²)+½(1)(4²) = 1 + 8 = 9 J ✓

Perfectly inelastic after: v = 6/(2+1) = 2 m/s. Check: p = (3)(2) = 6 kg·m/s ✓; KE = ½(3)(2²) = 6 J (3 J lost to heat/sound)

Results Table — Collision Trials

Type	p before (kg·m/s)	p after (kg·m/s)	KE before (J)	KE after (J)
Elastic	6.0	6.0	9.0	9.0
Inelastic (stick)	6.0	6.0	9.0	6.0
Elastic (equal mass)	4.0	4.0	8.0	8.0

In the equal-mass elastic head-on trial the moving ball stopped and transferred all its velocity to the struck ball, as predicted for equal masses.

Discussion

In every trial the total momentum after the collision equaled the total momentum before, confirming conservation of momentum regardless of collision type. The kinetic energy behaved differently: in the elastic trials the total kinetic energy after matched the value before (9.0 J before and after), but in the perfectly inelastic trial the kinetic energy dropped from 9.0 J to 6.0 J. The missing 3.0 J was converted to heat, sound, and permanent deformation when the balls stuck together.

The equal-mass elastic collision produced the familiar result that the incoming ball stops while the target ball moves off with the incoming ball's original velocity — a complete exchange of velocity that conserves both momentum and kinetic energy. These results confirm that momentum conservation is universal, while kinetic-energy conservation is the special signature of an elastic collision.

Conclusion

The experiment verified that total linear momentum is conserved in all collisions, elastic and inelastic alike, and that total kinetic energy is conserved only in elastic collisions. Perfectly inelastic collisions conserve momentum but lose the most kinetic energy, consistent with the conversion of mechanical energy into other forms.

Practice Questions

Show all work and include units in your answers.

Question 1

A 0.50 kg ball moving at 6.0 m/s strikes a wall and bounces straight back at 6.0 m/s. Find the change in the ball's momentum.

Hint: Δp = p_after − p_before = m(−6) − m(+6). Watch the signs.

Question 2

A 2.0 kg ball at 4.0 m/s collides and sticks to a 3.0 kg ball at rest. Find their common velocity and the kinetic energy lost in the collision.

Hint: v = (m₁u₁)/(m₁+m₂). Compute KE before and after; the difference is lost.

Question 3

A 1.0 kg ball at +5.0 m/s collides elastically with a 1.0 kg ball at rest. Find both final velocities.

Hint: for equal masses in an elastic head-on collision, the balls exchange velocities.

Question 4

Two ice skaters, 60 kg and 40 kg, stand at rest and push off from each other. The 60 kg skater moves at 2.0 m/s. How fast does the 40 kg skater move, and in which direction?

Hint: total momentum starts at zero, so m₁v₁ + m₂v₂ = 0. Solve for v₂ (opposite direction).

Question 5

A 0.10 kg arrow is shot at 50 m/s into a 2.4 kg block resting on a frictionless surface; the arrow embeds in the block. Find the speed of the block-and-arrow afterward.

Hint: perfectly inelastic — v = (m_arrow·u_arrow)/(m_arrow + m_block).

Question 6 — Challenge

A 3.0 kg ball at +4.0 m/s collides elastically with a 1.0 kg ball at rest. Use conservation of momentum and kinetic energy to find both final velocities, then verify that momentum and KE are both conserved.

Hint: v₁ = (m₁−m₂)u₁/(m₁+m₂) = (2)(4)/4 = 2 m/s; v₂ = 2m₁u₁/(m₁+m₂) = (24)/4 = 6 m/s. Check p and KE.