MechanicsIntro

Concept module

Momentum and Impulse

Push one cart with a timed force pulse and watch momentum, impulse, and force-time area stay tied to the same motion, readouts, and graphs.

Interactive lab

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

Time

0.00 s / 2.50 sLivePause to inspect a specific moment, then step or scrub through it.

0.00 s2.50 s

Momentum and Impulse

One cart, one timed force pulse, and three linked graphs are enough to keep momentum, impulse, and force-over-time honest without building a full collision sandbox.

Graphs

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

Force vs time

Shows the height, sign, and width of the force pulse.

time (s): 0 to 2.5force (N): -4 to 4

Applied force

Hover or scrub to link the graph back to the stage.time (s) / force (N)

Controls

Adjust the physical parameters and watch the motion respond.

Mass

m

1 kg

Change how much mass is moving without changing the force pulse.

Initial velocity

v_{i}

0.5 m/s

Set the cart's starting motion before the pulse begins.

Force

F

3 N

Positive force pushes right. Negative force pushes left.

Pulse duration

Δ t

0.4 s

Set how long the force stays on during the fixed pulse window.

More tools

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

Show

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 pulse area on the force graph is the same quantity that appears as the momentum change.

Try this

Watch the force pulse and the accumulated-impulse graph together. The momentum graph should shift by the same signed amount.

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

Mass

1 kg

Heavier carts need more impulse for the same velocity change, even though the momentum change still follows the same force-time area.

Graph: Momentum vs timeOverlay: 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 the pulse. The best prompt should point at something the stage and the graph can both honestly show right now.

ObservationPrompt 1 of 1

Graph: Force vs time

Notice that the pulse area on the force graph is the same quantity that appears as the momentum change.

Try this

Watch the force pulse and the accumulated-impulse graph together. The momentum graph should shift by the same signed amount.

Why it matters

This is the impulse-momentum theorem in one bounded visual loop.

Control: ForceControl: Pulse durationGraph: Force vs timeGraph: Impulse and change in momentumGraph: Momentum vs timeOverlay: Pulse windowOverlay: Momentum barsEquation

F

Equation

Δ t

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

Force arrow

Shows the direction and size of the applied force on the cart.

What to notice

The arrow only matters during the pulse window. Before and after that, the cart coasts.
Reversing the arrow reverses the sign of the impulse and the momentum change.

Why it matters

Impulse is signed, so direction matters just as much as size.

Control: ForceGraph: Force vs timeGraph: Impulse and change in momentumEquation

F

At t = 0 s, the 1 kg cart is moving to the right at 0.5 m/s. Its momentum is 0.5 kg m/s, and the pulse has delivered 0 N s of impulse so far. The pulse has not started yet, so momentum still matches the initial value.

Equation detailsDeeper interpretation, notes, and worked variable context.

Momentum

Momentum combines how much mass is moving with how fast it moves.

p = m v

m

Mass 1 kg

v_{i}

Initial velocity 0.5 m/s

Impulse for a constant pulse

For a constant force, impulse is the force-time area.

J = F Δ t

F

Force 3 N

Δ t

Pulse duration 0.4 s

Impulse-momentum theorem

Impulse changes momentum directly, which is why the accumulated impulse and momentum shift should match.

J = Δ p = p_{f} - p_{i}

m

Mass 1 kg

v_{i}

Initial velocity 0.5 m/s

F

Force 3 N

Δ t

Pulse duration 0.4 s

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

Momentum tells you how much motion an object is carrying: in one dimension it is $p = m v$ , so a heavier cart or a faster cart has more momentum. The sign matters too, because momentum points with the velocity.

Impulse is what a force does over a stretch of time. In this module one bounded force pulse pushes a cart on a fixed track, so you can see the pulse shape, the accumulated impulse, and the momentum graph move together without inventing a separate subsystem for collisions.

Key ideas

01Momentum is mass times velocity, so the same speed change does not mean the same momentum change for every mass.

02Impulse is the signed force-time area. A short large force and a smaller longer force can produce the same total impulse if the areas match.

03For the same cart, impulse changes momentum directly through $J = \Delta p$, which is why the momentum graph and the accumulated-impulse graph should agree.

Live impulse checks

Solve the exact state on screen.

Solve the pulse you are actually watching. Time-based examples follow the current inspected time unless you freeze them, and the pulse example keeps the full force-duration pair tied to the real controls.

Live valuesLive at t = 0.00 s

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

t

Time

0 s

m

Mass

1 kg

v

Velocity

0.5 m/s

1. Identify the relation

Use

p = m v

for the cart at the currently inspected moment.

2. Substitute the live values

p = 1 \cdot 0.5

3. Compute the momentum

That gives

p = 0.5 kg m/s

Current momentum

p = 0.5 kg m/s

Before the pulse starts, the cart keeps its initial velocity, so the current momentum still matches the starting value.

Force-pulse checkpoint

Can you choose a new force-duration pair that leaves the final momentum unchanged?

Prediction prompt

Try making the pulse shorter while raising the force, or longer while lowering the force. Predict which pairs should keep the same final momentum.

Check your reasoning

Any pair with the same signed area under the force-time graph gives the same momentum change for the same cart.

This is the bounded version of the impulse idea. If

F Δ t

stays the same, then

J

stays the same, so the momentum change stays the same even though the pulse shape looks different.

Common misconception

A bigger force always means a bigger momentum change, even if it acts for less time.

Momentum change depends on the total impulse, not force alone. A large force over a very short interval can match the effect of a smaller force acting longer.

Mass changes how much the velocity shifts for a given impulse, but it does not change the fact that the momentum change itself is set by $J = F Δ t$ .

Quick test

Compare cases

Question 1 of 4

Use the force pulse, the cart motion, and the graphs together. Each question checks whether you can reason with momentum and impulse, not just recite a definition.

A $3 N$ force for $0.4 s$ and a $1.5 N$ force for $0.8 s$ act on the same cart in the same direction. Which statement is correct?

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 a cart moving on a fixed horizontal track while a single force pulse turns on during a highlighted time window. The cart label shows mass, a horizontal arrow shows velocity, and overlays can emphasize the force direction, the pulse window, and centered bars for initial, current, and final momentum.

Changing mass, initial velocity, force, or pulse duration immediately updates the cart motion, the pulse timing, the readouts, and the linked graphs without changing the underlying scale.

Graph summary

The force graph shows a rectangular pulse that can point positive or negative. The momentum graph is flat before and after the pulse and changes only while the force acts.

The impulse graph and the change-in-momentum graph should overlap because they represent the same signed quantity for the same cart.

Keep mechanics 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.

Two-object systemsMechanics

Momentum and Impulse

See each variable before you move it.

Force arrow

What the system is doing

Solve the exact state on screen.

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

A $3 N$ force for $0.4 s$ and a $1.5 N$ force for $0.8 s$ act on the same cart in the same direction. Which statement is correct?

Keep moving through the concept map.

Conservation of Momentum

Collisions

Projectile Motion

Momentum and Impulse

See each variable before you move it.

Force arrow

What the system is doing

Solve the exact state on screen.

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

A 3N force for 0.4s and a 1.5N force for 0.8s act on the same cart in the same direction. Which statement is correct?

Keep moving through the concept map.

Conservation of Momentum

Collisions

Projectile Motion

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

A $3 N$ force for $0.4 s$ and a $1.5 N$ force for $0.8 s$ act on the same cart in the same direction. Which statement is correct?