Classical Mechanics · Concept #3Momentum & Impulse 5 min read

Conservation of Momentum & Elastic vs. Inelastic Collisions

Q: What is impulse (J = F·Δt) and how does it relate to momentum?

Impulse is just the change in momentum (Δp = F × Δt). To change an object’s momentum by a fixed amount, you can either apply a huge force for a tiny split second (like a cue stick striking a hard ball) or a gentle force stretched over a longer time (like an airbag or the pneumatic dampener in Celestial Harvest).

Why a cue ball stops dead on a head-on hit, why angled billiard shots split at 90°, and what actually changes between bouncy and sticky crashes.

Core Formulap = m · v | m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂ | e = |v₂ - v₁| / |u₁ - u₂|

Total Momentum Before Crash = Total Momentum After Crash (Always—Even in Sticky Crashes!)

Quick Answer in 30 Seconds

Momentum (p = m·v) is a measure of how hard it is to stop a moving object. The Law of Conservation of Momentum states that when two objects collide—with no outside forces like heavy friction interfering—the total momentum arrow of the system after the crash equals the exact total momentum arrow before the crash. That rule holds true whether the objects bounce apart cleanly (elastic) or mash together into one lump (inelastic).

Momentum is a vector (it cares about direction: left is negative, right is positive!).
Momentum is ALWAYS conserved in isolated collisions—both elastic and inelastic.
Kinetic Energy (½mv²) is only conserved in 100% bouncy ELASTIC collisions (e = 1.0).

2D Collision Physics Arcade

Don't Just Read the Math — Play It

Jump straight into Momentum Clash: Newtonian Billiards in your browser (zero install) and test this exact physical law at 60 FPS:

Play Momentum Clash Live

Step 1 · Direction Matters

Why Momentum Is an Arrow (And Why the 90° Billiards Trick Works)

If a 1 kg ball rolls East at 5 m/s and an identical 1 kg ball rolls West at 5 m/s, what is the total momentum of the pair? It is not 10—it is ZERO! Because momentum is a vector (+5 + (−5) = 0), opposite directions cancel out.

That vector rule explains the coolest trick in pool and in Momentum Clash. Suppose you shoot a cue puck straight forward into a stationary target puck of the exact same mass, clipping it slightly on the left.

Which way does the target puck go? It ALWAYS shoots along the straight line connecting the two centers at the moment they touch. No exceptions!

And what about your cue puck? Because the target puck gained some upward sideways momentum (+y), your cue puck MUST gain the exact same amount of downward sideways momentum (−y) so the vertical total stays at zero. If the collision is elastic and neither ball has spin, the two pucks fly apart at an exact 90-degree right angle!

Sketch diagram of a 2D billiards collision showing the line of centers, the 90-degree split between equal-mass pucks, and the tip-to-tail momentum vector triangle. — Sketch 3.1: The 2D Billiards Split & Tip-to-Tail Vector Triangle
Every pool shark uses this instinctively: draw an imaginary line through the centers of the two balls at impact to find where the target goes, then look 90° away to see where your cue ball will roll.
① Target puck always travels along the line connecting the two centers at impact.② Upward (+y) and downward (−y) sideways momentum cancel out to zero.③ For equal masses in an elastic hit, the two outgoing paths form a 90° angle.

Step 2 · Elastic vs. Inelastic

Bouncy vs. Sticky Crashes: What Stays and What Disappears?

Physics teachers love asking a classic trap question on tests: "Two lumps of clay smash together and stick. Is momentum conserved, or is kinetic energy conserved?"

Here is the clean answer: Momentum is ALWAYS conserved in both elastic and inelastic crashes (as long as no external wall or heavy friction pushes on them during the brief impact). Why? Because of Newton’s Third Law! For every millisecond that Puck A pushes forward on Puck B (+F), Puck B pushes backward on Puck A with the exact same force (−F) for the exact same time (Δt). Whatever momentum A loses, B gains.

Kinetic energy (½mv²) is different. When two hard billiard pucks collide elastically (e = 1), they flex like tiny springs and snap right back out, returning 100% of the motion energy. When two soft clay blobs or cars crunch together (inelastic, e < 1), part of that motion energy goes into permanently bending metal, squishing clay, and making heat and sound.

Two-panel sketch comparing an elastic collision where a moving puck stops dead and transfers 10 m/s to the target, versus an inelastic collision where both stick together at 5 m/s. — Sketch 3.2: Side-by-Side Comparison — Elastic (e = 1) vs. Perfectly Inelastic (e = 0)
Notice in Panel B: when two equal masses stick together, mass doubles (2m) so speed drops in half (5 m/s). Momentum (2m × 5 = 10m) is 100% intact, while half the kinetic energy turned into heat!
Elastic (e = 1): Equal-mass head-on hit trades velocities cleanly (Cue stops, Target takes 100% speed).Inelastic (e = 0): Pucks lock together at half speed; momentum is still 100% conserved!

Step 3 · Mass Ratios

What Happens When a Light Puck Hits a Heavy Tank (Or Vice Versa)?

In Momentum Clash, you do not just shoot equal-weight pucks—you also face heavy high-mass blockers and tiny lightweight targets. How does mass ratio change a head-on elastic bounce?

1. Equal Mass (m₁ = m₂, like two standard pool balls): The cue puck stops dead in its tracks (v₁ = 0), handing 100% of its velocity to the target puck.

2. Light Hits Heavy (m₁ < m₂, like a ping-pong ball hitting a bowling ball): The heavy puck barely budges forward, and your light cue puck rebounds BACKWARD in the direction it came from!

3. Heavy Hits Light (m₁ > m₂, like a bowling ball hitting a ping-pong ball): The heavy cue puck keeps plowing forward almost as fast as before, while the tiny target puck shoots ahead at nearly DOUBLE the cue puck’s speed!

Formula Breakdown

The Collision Accounting Equation

Think of momentum like cash in a closed room between two people. Puck 1 can hand some (or all) of its momentum over to Puck 2 during a smack, but the total amount in the room never changes:

m₁·u₁ + m₂·u₂ = m₁·v₁ + m₂·v₂

Symbol	What It's Called	Plain-English Meaning	How It Feels in Momentum Clash: Newtonian Billiards
`p = m · v`kg·m/s (or N·s)	Linear Momentum	Mass times velocity. A slow freight train and a fast baseball can carry the same momentum.	Shown by the length and direction of the glowing velocity/momentum arrows on each puck.
`u₁, u₂`m/s (with +/− sign!)	Initial Velocities (Before)	How fast and which way each puck is moving right before they touch.	Controlled by how far back you pull your cue shot.
`v₁, v₂`m/s (with +/− sign!)	Final Velocities (After)	How fast and which way each puck moves right after separating.	Determines whether your target puck drops into the pocket or your cue scratches.
`e (0 to 1)`unitless (0 to 1)	Coefficient of Restitution	The "bounciness" score: separation speed divided by approach speed. e = 1 is perfectly bouncy; e = 0 sticks together.	High-e bumpers launch pucks at full speed; low-e dampeners bleed off kinetic energy.

Try It With Real Numbers

Quick Math Check: Heavy Puck vs. Light Puck Head-On

Scenario: In Momentum Clash, you fire a heavy 3 kg Striker puck moving right at u₁ = +8 m/s head-on into a stationary 1 kg Scout puck (u₂ = 0 m/s). The collision is perfectly elastic (e = 1).

Step 1: Calculate total momentum BEFORE impact

p_total = (3 kg × 8 m/s) + (1 kg × 0 m/s) = +24 kg·m/s

No matter what happens next, the final momentum of the two pucks added together must equal +24 kg·m/s.

Step 2: Use the elastic head-on velocity formulas

v₁ = [(m₁ - m₂)/(m₁ + m₂)] · u₁ = (2/4) × 8 = +4 m/s   |   v₂ = [2m₁/(m₁ + m₂)] · u₁ = (6/4) × 8 = +12 m/s

The heavy 3 kg puck slows down from +8 m/s to +4 m/s (still moving forward!), while the light 1 kg puck rockets ahead at +12 m/s.

Step 3: Double-check that momentum AND separation speed match

p_after = (3 × 4) + (1 × 12) = 12 + 12 = +24 kg·m/s ✔ | v₂ - v₁ = 12 - 4 = 8 m/s ✔

Total momentum is still 24 kg·m/s, and the speed at which they separate (8 m/s) matches the speed at which they approached (8 m/s).

Gameplay Tip: When you need a stationary target puck to fly across the arena at maximum speed in Momentum Clash, hit it with a heavier cue puck!

Common Questions

Frequently Asked Questions

Is momentum conserved when a ball bounces off a wall or cushion?

The ball’s momentum reverses sign (for example, from +5 to −5 kg·m/s), so the ball alone changes momentum by −10 kg·m/s. Where did the other +10 kg·m/s go? Into the wall and the Earth attached to it! Because the Earth weighs 6 × 10²⁴ kg, its recoil speed is unimaginably tiny, so total momentum of (Ball + Earth) is still conserved.

Why does a cue ball sometimes follow forward instead of stopping dead on a straight shot?

On a real pool table, a rolling ball has topspin (angular momentum). Even after a head-on collision stops its forward sliding motion for a split second, its spinning grip on the felt cloth pulls it forward again. In a frictionless slide (or stun shot), it stops completely dead.

How do you tell on a physics test if a collision is elastic or inelastic?

Check whether the objects stick together (always inelastic) or calculate ½m₁v₁² + ½m₂v₂² before and after. If total kinetic energy after equals total kinetic energy before, it is elastic (e = 1). If kinetic energy drops, it is inelastic (e < 1).

What is impulse (J = F·Δt) and how does it relate to momentum?