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MechanicsIntermediateStarter track

Concept module

Escape Velocity

Launch outward from one bounded gravity source and see how source mass, launch radius, and total specific energy decide whether the object escapes or eventually returns.

Interactive lab

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

Time

0.00 s / 5.45 sLivePause to inspect a specific moment, then step or scrub through it.

0.00 s5.45 s

Escape Velocity

Launch outward from one chosen radius and keep the trajectory, threshold speeds, and specific energies on the same live state so escape stays an energy question instead of a distance guess.

Graphs

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

Radius and launch radius

Shows how far the launch has traveled compared with the starting radius.

time (s): 0 to 5.45radius (m): 0 to 9

r(t)r_launch

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

Controls

Adjust the physical parameters and watch the motion respond.

Source mass

M

4 kg

Changes the source mass creating the well and therefore the escape threshold.

Launch radius

r_{0}

1.6 m

Sets how far from the source the outward launch begins.

Speed factor

df r a c v_{0} v_{t e x t esc}

1 x escape

1.00 means the launch speed exactly matches the escape threshold for the current source mass and launch radius.

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.

Try thisPrompt 1 of 4

Launching farther from the source lowers the threshold speed because the well is shallower at the starting point, not because gravity has disappeared.

Try this

Use Wider launch threshold and compare the new launch marker with the lower threshold speed and the less-negative starting potential energy.

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

Source mass

4 kg

Changes the depth of the gravity well, so the escape threshold, circular comparison speed, and gravity pull all scale from the same cause.

Graph: Actual speed and threshold speedsGraph: Specific energy balanceOverlay: Gravity vectorOverlay: Velocity vectorOverlay: Trajectory trail

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 one prompt at a time. The main pattern changes depending on whether you inspect the energy line, change the source mass or launch radius, or compare a bound launch with a true escape.

Try thisPrompt 1 of 4

Graph: Radius and launch radius

Launching farther from the source lowers the threshold speed because the well is shallower at the starting point, not because gravity has disappeared.

Try this

Use Wider launch threshold and compare the new launch marker with the lower threshold speed and the less-negative starting potential energy.

Why it matters

It makes the radius dependence feel like an energy-depth story instead of a formula to memorize.

Control: Launch radiusGraph: Radius and launch radiusGraph: Actual speed and threshold speedsGraph: Specific energy balanceOverlay: Launch markerOverlay: Trajectory trailEquation

r_{0}

Guided overlays

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

5 visible

Overlay focus

Launch marker

Marks the starting radius so the equations and trail stay tied to the same launch point.

What to notice

Changing only the launch radius moves the start line and immediately changes the threshold speed.

Why it matters

It keeps r_0 visible as the launch condition.

Control: Launch radiusGraph: Radius and launch radiusGraph: Actual speed and threshold speedsEquation

r_{0}

Challenge mode

Tune the same one-source launch bench into threshold targets. The checks read the live energy, speed, and turnaround state instead of a detached worksheet.

0/2 solved

TargetCore

7 of 8 checks

Remove the finite turnaround

Starting from High but bound, raise the launch just to the threshold case at the same source mass and launch radius so the total specific energy is about zero and the finite turnaround disappears.

Graph-linkedGuided start2 hints

Suggested start

Keep the same source mass and launch radius. Only the launch speed should move.

Pending

Open the Specific energy balance graph.

Radius and launch radius

Matched

Keep the Turnaround marker visible.

Matched

Keep the Trajectory trail visible.

Matched

Keep source mass between 3.95 kg and 4.05 kg.

4 kg

Matched

Keep launch radius between 1.55 m and 1.65 m.

1.6 m

Matched

Keep speed factor between 0.99 and 1.01.

Matched

Keep total energy between -0.02 and 0.02.

4.44e-16

Matched

Keep launch speed between 2.22 m/s and 2.25 m/s.

2.24 m/s

The checklist updates from the live simulation state, active graph, overlays, inspect time, and compare setup.

At t = 0 s, the launch mass is 1.6 m from the source and moving at 2.24 m/s along the radial line. The local escape speed there is 2.24 m/s, while the circular-orbit comparison speed at that same radius is 1.58 m/s. The specific energies are K/m = 2.5, U/m = -2.5, and E/m = 4.44e-16. The launch sits close to the escape threshold, where the total specific energy is about zero and the speed slowly trends toward zero only very far away. There is no finite turnaround radius because the total specific energy is zero or positive.

Equation detailsDeeper interpretation, notes, and worked variable context.

Specific total energy

The sign of the total energy per unit mass decides whether the launch is bound, threshold, or escaping.

df r a c E m = df r a c v^{2} 2 - df r a c M r

M

Source mass 4 kg

df r a c v_{0} v_{t e x t esc}

Speed factor 1 x escape

Threshold condition

Escape velocity is the launch speed that makes the total specific energy exactly zero at the starting radius.

0 = df r a c v_{t e x t esc}^{2} 2 - df r a c M r_{0}

M

Source mass 4 kg

r_{0}

Launch radius 1.6 m

df r a c v_{0} v_{t e x t esc}

Speed factor 1 x escape

Escape speed

The threshold rises for heavier source mass and falls for larger launch radius.

v_{t e x t esc} = s q r t df r a c 2 M r_{0}

M

Source mass 4 kg

r_{0}

Launch radius 1.6 m

Circular-speed comparison

Local circular speed is useful for contrast, but exceeding it alone does not guarantee escape.

v_{c} = s q r t df r a c M r_{0}, q q u a d v_{t e x t esc} = s q r t 2, v_{c}

M

Source mass 4 kg

r_{0}

Launch radius 1.6 m

Finite turnaround radius

A negative total specific energy predicts the largest radius before the bound launch must reverse.

r_{t e x t t u r n} = df r a c M - (E / m) q q u a d (E < 0)

M

Source mass 4 kg

r_{0}

Launch radius 1.6 m

df r a c v_{0} v_{t e x t esc}

Speed factor 1 x escape

Progress

Not startedMastery: NewLocal-first

Start exploring and Open Model Lab will keep this concept's progress on this browser first. Challenge mode has 2 compact tasks ready. 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

Starter track

Step 5 of 50 / 5 complete

Gravity and Orbits

Earlier steps still set up Escape Velocity.

1. Gravitational Fields2. Gravitational Potential and Potential Energy3. Circular Orbits and Orbital Speed4. Kepler's Third Law and Orbital Periods+1 more steps

Previous step: Kepler's Third Law and Orbital Periods.

Start from track beginning

Short explanation

What the system is doing

Escape velocity is the minimum outward launch speed from one chosen radius that makes the total specific energy reach zero. Above that threshold, gravity still pulls inward and slows the launch, but there is no finite turnaround radius. Below it, even a very high outward trip is still bound and eventually returns.

This bounded lab keeps one source mass, one launch radius, one speed factor, one live radial path, and the linked radius, speed-threshold, and specific-energy graphs on the same state. The local circular-speed comparison stays visible too, so going far away is not confused with escape or with circular orbit balance.

Key ideas

01Escape is an energy threshold, not a place where gravity turns off.

02With displayed units using G = 1, the escape speed from radius r_0 is v_esc = sqrt(2M/r_0), so a heavier source raises the threshold while a larger launch radius lowers it.

03A launch can be faster than the local circular speed and still remain bound if it stays below escape speed and keeps negative total specific energy.

Live escape checks

Solve the exact state on screen.

Use the same source mass, launch radius, speed factor, live time, and graphs already on screen.

Live valuesFollowing current parameters

For the current source mass and launch radius, what outward launch speed makes the total specific energy exactly zero?

M

Source mass

4 kg

r_{0}

Launch radius

1.6 m

1. Set the threshold condition

At escape threshold,

0 = df r a c v_{t e x t esc}^{2} 2 - df r a c M r_{0}

2. Solve for the threshold speed

v_{t e x t esc} = s q r t 2 M / r_{0} = s q r t 2 (4) /1.6

in the displayed

G = 1

units.

3. Compute the live value

That gives

v_{t e x t esc} = 2.24, ma t h r m m / s

, while the circular-speed comparison is

v_{c} = 1.58, ma t h r m m / s

at the same radius.

Escape speed

v_{t e x t esc} = 2.24, ma t h r m m / s

The escape threshold comes straight from setting the total energy to zero at the launch point.

Escape-threshold checkpoint

Start from High but bound. Without changing the source mass or launch radius, what single control change removes the finite turnaround radius?

Prediction prompt

Decide what should happen to the total-energy line and the turnaround marker when the launch just reaches escape threshold.

Check your reasoning

Raise the launch speed until the speed factor reaches 1.00, so the total specific energy reaches zero and the finite turnaround radius disappears.

At fixed source mass and launch radius, changing the speed factor changes the kinetic term at launch. The threshold case happens when

E / m = v_{0}^{2} /2 - M / r_{0} = 0

Common misconception

If a launch gets far enough from the source, it has escaped, and escape means gravity is basically gone.

Distance alone does not decide escape. The deciding quantity is the sign of E/m. Zero or positive total specific energy means there is no finite turnaround radius; negative total specific energy means the launch is still bound.

Gravity does not vanish after escape. It keeps acting and keeps reducing the speed. The difference is that the object never has to reverse direction.

Quick test

Reasoning

Question 1 of 4

Answer from the live launch-and-energy logic, not from detached formulas.

A launch begins faster than the local circular-speed comparison but still below the escape speed from the same radius. What must be true in this lab?

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 one source mass on the left side of a compact radial launch axis, a launched mass moving outward or inward along that axis, and optional overlays for the launch marker, finite turnaround marker, current velocity vector, inward gravity vector, and visited trajectory trail.

Changing source mass, launch radius, or speed factor updates the same trajectory, readout card, and linked graphs together. Compare mode overlays a second launch on a separate dashed track instead of switching to a different model.

The displayed units use a bounded one-source gravity model with G = 1. The stage has a finite maximum visible radius, and bound launches whose turnaround sits beyond that view are labeled explicitly rather than being faked into the visible window.

Graph summary

The radius-history graph compares the live radius with the starting launch radius over the same time window. Hovering or scrubbing the graph previews the same instant on the launch stage.

The speed-thresholds graph compares the live speed with the local escape-speed and circular-speed benchmarks, and the specific-energy graph shows the live kinetic, potential, and total specific energies together with the zero-energy threshold.

Keep the gravity launch 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.

Bound-orbit timingMechanics