MechanicsIntro

Concept module

Collisions

Collide two carts on one honest track, keep total momentum in view, and see how elasticity, mass, and incoming speed shape the rebound or stick-together outcome.

Interactive lab

Keep the stage, graph, and immediate control feedback in one working view.

Time

0.00 s / 4.50 sLivePause to inspect a specific moment, then step or scrub through it.

0.00 s4.50 s

Collisions

Two carts collide on one fixed track. Total momentum stays visible, elasticity controls how much closing speed comes back as rebound, and the energy graph distinguishes elastic from inelastic outcomes without leaving the same live state.

Graphs

Switch graph views without breaking the live stage and time link.

Velocities and center-of-mass speed

Shows the velocity jump at contact and the steady center-of-mass speed that carries the system total momentum.

time (s): 0 to 4.5velocity (m/s): -2 to 2

Velocity of AVelocity of BCenter-of-mass velocity

Hover or scrub to link the graph back to the stage.time (s) / velocity (m/s)

Controls

Adjust the physical parameters and watch the motion respond.

Mass of A

m_{A}

1.2 kg

Change cart A without changing the track or the collision point.

Mass of B

m_{B}

2.2 kg

Change cart B and compare how its velocity response differs from A's.

Incoming speed of A

u_{A}

1.6 m/s

Set how quickly cart A approaches from the left.

Incoming speed of B

u_{B}

0.7 m/s

Set how quickly cart B approaches from the right. Zero means B starts at rest.

More tools

Secondary controls, alternate presets, and less-used toggles stay nearby without crowding the main bench.

Show

Elasticity

e

0.8

Zero means perfectly inelastic. One means perfectly elastic.

More presets

Presets

Predict -> manipulate -> observe

Keep the active prompt next to the controls so each change has an immediate visible consequence.

ObservationPrompt 1 of 1

Notice that the center-of-mass marker keeps one steady drift even though the individual cart velocities jump at contact.

Try this

Leave the center-of-mass overlay on while you scrub through contact. The CM marker should not kink or bounce.

Equation map

See each variable before you move it.

Select a symbol to highlight the matching control and the graph or overlay it most directly changes.

m_{A}

Mass of A

1.2 kg

Changing A's mass changes how strongly a given momentum change shifts A's velocity after contact.

Graph: Velocities and center-of-mass speedGraph: Object and total momentumOverlay: Momentum bars

Equations in play

Choose an equation to sync the active symbol, control highlight, and related graph mapping.

More tools

Detailed noticing prompts, guided overlays, and challenge tasks stay available without taking over the main bench.

Hide

What to notice

Use the live prompt as a short guide while you change masses, incoming speeds, and elasticity. The best prompt should point at something the stage and the current graph already show honestly.

ObservationPrompt 1 of 1

Graph: Velocities and center-of-mass speed

Notice that the center-of-mass marker keeps one steady drift even though the individual cart velocities jump at contact.

Try this

Leave the center-of-mass overlay on while you scrub through contact. The CM marker should not kink or bounce.

Why it matters

A steady center-of-mass motion is another honest sign that the system total momentum has not changed.

Graph: Velocities and center-of-mass speedGraph: Object and total momentumOverlay: Center of mass

Guided overlays

Focus one overlay at a time to see what it represents and what to notice in the live motion.

3 visible

Overlay focus

Collision zone

Marks the contact point on the track and the contact instant on the timeline.

What to notice

The sudden velocity change happens only at the contact instant, not throughout the whole run.
The collision point stays fixed, so compare mode never cheats by moving the target just to fit the picture.

Why it matters

A short collision is still a real physical event in time, and the graphs should jump at the same instant the stage reaches contact.

Control: Incoming speed of AControl: Incoming speed of BGraph: Velocities and center-of-mass speedGraph: Object and total momentumGraph: Kinetic energy through the collisionEquation

u_{A}

Equation

u_{B}

At t = 0 s, cart A has 1.92 kg m/s and cart B has -1.54 kg m/s. The total momentum is 0.38 kg m/s while the total kinetic energy is 2.08 J. Before contact, cart A approaches at 1.6 m/s while cart B approaches at -0.7 m/s.

Equation detailsDeeper interpretation, notes, and worked variable context.

Momentum of one cart

Each cart's momentum comes from its own mass and velocity.

p = m v

m_{A}

Mass of A 1.2 kg

m_{B}

Mass of B 2.2 kg

Momentum conservation across contact

The system total momentum before the collision must equal the system total momentum after the collision.

m_{A} u_{A} + m_{B} u_{B} = m_{A} v_{A} + m_{B} v_{B}

m_{A}

Mass of A 1.2 kg

m_{B}

Mass of B 2.2 kg

u_{A}

Incoming speed of A 1.6 m/s

u_{B}

Incoming speed of B 0.7 m/s

Elasticity relation

Elasticity compares the rebound speed after contact with the closing speed before contact.

v_{B} - v_{A} = e (u_{A} - u_{B})

u_{A}

Incoming speed of A 1.6 m/s

u_{B}

Incoming speed of B 0.7 m/s

e

Elasticity 0.8

Kinetic energy

Kinetic energy helps distinguish elastic collisions from inelastic ones because only elastic collisions keep the total kinetic energy unchanged.

K = \frac{1}{2} m v^{2}

e

Elasticity 0.8

Progress

Not startedMastery: NewLocal-first

Start exploring and Open Model Lab will keep this concept's progress on this browser first. No finished quick test, solved challenge, or completion mark is saved yet.

Let the live model runChange one real controlOpen What to notice

Try this setup

Copy the live bench state and reopen this concept with the same controls, graph, overlays, and compare context.

Stable links

Short explanation

What the system is doing

A collision is a short interaction, but it does not erase the momentum story. If the two-cart system is isolated, then the total momentum before contact must match the total momentum after contact even when the individual cart velocities change abruptly.

This module keeps the setup bounded and honest with two carts on one fixed track, one contact point, and one elasticity control. You can change the masses and incoming speeds, then compare an elastic rebound with a more inelastic outcome without turning the page into a giant sandbox.

Key ideas

01Momentum conservation belongs to the whole two-cart system, so the total momentum line stays flat through contact.

02Elasticity tells you how much of the closing speed comes back as separation speed. Elastic collisions keep kinetic energy, while inelastic ones lose some kinetic energy even though momentum is still conserved.

03The same momentum rule can give very different post-collision speeds because mass changes how a given momentum change affects velocity.

Live collision checks

Solve the exact state on screen.

Solve the collision you are actually watching. The total-momentum check follows the current inspected moment, while the outcome check uses the current masses, incoming speeds, and elasticity to predict the post-collision velocities.

Live valuesLive at t = 0.00 s

At $t = 0 s$ , what is the system's total momentum?

m_{A}

Mass of A

1.2 kg

v_{A}

Velocity of A

1.6 m/s

m_{B}

Mass of B

2.2 kg

v_{B}

Velocity of B

-0.7 m/s

1. Add the cart momenta

Use

p_{tot} = m_{A} v_{A} + m_{B} v_{B}

for the same instant shown on the stage.

2. Substitute the live values

p_{tot} = (1.2) (1.6) + (2.2) (- 0.7)

3. Compute the total

p_{tot} = 0.38 kg m/s

Current total momentum

p_{tot} = 0.38 kg m/s

Before contact, the system total is already fixed by the incoming masses and velocities, so the collision has not created a new total.

Collision outcome checkpoint

Can you make cart A bounce backward while the system total momentum still points to the right?

Prediction prompt

Try a light cart A hitting a heavier cart B with a fairly elastic rebound. Predict whether the heavy cart can carry enough rightward momentum to keep the total positive.

Check your reasoning

Yes. Cart A can reverse while the total momentum stays rightward if cart B leaves the collision with enough positive momentum.

Momentum conservation constrains the sum, not the direction of each object. A light cart can rebound left after contact while the heavier cart keeps the system total moving right.

Common misconception

If momentum is conserved, then the carts must keep the same total kinetic energy and rebound the same way every time.

Momentum conservation does not guarantee kinetic-energy conservation. Inelastic collisions can keep the same total momentum while the total kinetic energy drops.

Mass and incoming velocity still matter. A light cart hitting a heavy one can rebound sharply, while a heavy cart can keep moving forward with only a modest speed change.

Quick test

Reasoning

Question 1 of 4

Use the stage and the linked graphs together. These checks are about reasoning through collisions, not just repeating the word conservation.

Which quantity must stay the same for this isolated two-cart collision system?

Choose one answer to reveal feedback, then test the idea in the live system if a guided example is available.

Accessible description

The simulation shows two carts moving on a fixed horizontal track toward one collision point. Each cart has a mass label and a horizontal velocity arrow, and optional overlays can mark the collision zone, the center of mass, the momentum bars, and the relative speed before or after contact.

Changing mass, incoming speed, or elasticity updates the same stage, readouts, and linked graphs without changing the track scale. A perfectly inelastic collision shows the carts leaving together, while higher elasticity shows them separating more strongly after contact.

Graph summary

The velocity graph shows the cart velocities changing at contact and includes a steady center-of-mass velocity line. The momentum graph shows the cart momentum lines changing while the total momentum line stays flat.

The energy graph shows when total kinetic energy is preserved and when it drops at contact, which is the main visual difference between elastic and inelastic behavior in this module.

Keep the momentum story moving

Keep moving through the concept map.

These suggestions come from the concept registry, so the reason label reflects either curated guidance or the fallback progression logic.

Apply momentum ideasMechanics

Collisions

See each variable before you move it.

Collision zone

What the system is doing

Solve the exact state on screen.

At $t = 0 s$ , what is the system's total momentum?

Which quantity must stay the same for this isolated two-cart collision system?

Keep moving through the concept map.

Projectile Motion

Momentum and Impulse

Conservation of Momentum

Collisions

See each variable before you move it.

Collision zone

What the system is doing

Solve the exact state on screen.

At t=0s, what is the system's total momentum?

Which quantity must stay the same for this isolated two-cart collision system?

Keep moving through the concept map.

Projectile Motion

Momentum and Impulse

Conservation of Momentum

At $t = 0 s$ , what is the system's total momentum?