Motion in a Potential

Source

Lecture: Classical mechanics — energy methods: 1-D motion in a potential landscape V(x)
Simulation: "Motion in a Potential — Bound, Free, or Balanced on the Edge" — energy-diagram mode with a ✓/⚠ counterfactual pair of speed rules
Concept: The height of the total-energy line E relative to the barrier top V_b — not the local slope — decides the fate of the motion: bound oscillation (E < V_b), escape (E > V_b), or the separatrix (E = V_b), where the time to reach the unstable maximum diverges.
Serves: Reading energy diagrams: allowed region, turning points where V(x)=E, KE = E − V(x); classifying equilibria via V″ (stable minimum at x = +1 m, unstable maximum at x = −1 m, barrier V_b = ⅔·s = 0.6667 J at s = 1); critical slowing on the separatrix.
Inquiry: Where along the track is the ball fastest? What changes as E crosses V_b? With E = V_b exactly, does the ball ever arrive at the top?
Misconception: "The ball's speed at a point is set by how steep the potential is there." Slope sets ACCELERATION a = −V′(x)/m; SPEED is set by E − V(x) — fastest at the minimum, zero at turning points.

Pedagogy principles applied

ID	Principle	How it shows up here
A1	One concept	One sentence: energy height (not slope) decides speed and fate; one hero energy diagram carries it.
A3/A4	Formal tools off by default	Equations (cubic V, v(x), separatrix log-divergence) live behind ⚛︎ The Physics — a pop-up revision sheet; the shell still loads with hide-formal.
C1/C2	Numbers earn their place	Only E and v (hero badge) are visible by default; the stopwatch t appears only for the edge/escape regimes; all else is in the Measurements panel.
D6/D7	One hero scene, staged extras	Speed-map v(x) and Measurements are side panels opened on demand; nothing is gated.
G4b	Explicit reveal controls	Bound/escape buttons (E below / above V_b), the Snap E = V_b separatrix control, and draggable E-line drive the reveal at the student's pace.
H1/H2	Misconception structural	"Steepest spot" is a prediction CHOICE; its rebuttal appears only when picked. ⚠ mode renders the wrong rule as a dashed amber ball that visibly stalls at the bottom.
—	Counterfactual mode	✓ v = √(2(E−V)/m) vs ⚠ v = (τ/m)\|V′\| (τ = 1 s, labeled) — both implemented exactly; the ⚠ ball ignores E, never turns, stalls at flat points.
E1/J2/J5	Honest animation, stable axes	Separatrix approach is a true exponential creep (t counts on; the ball never arrives); escape is one-pass; world axes are fixed — no auto-rescaling.
—	Real values / exact claims	SI units (J, m, m/s, m/s²; m = 1.00 kg; 1× real time); feedback numbers, t½ = ln2·√(m/2s), and regime thresholds computed from the sim's own formulas; window.__audit exposes speedAtMin = √(2(E−V_min)/m).
—	Robust at extremes	E is floored at V_min (rest at the stable equilibrium); the start point auto-snaps into the allowed region with an on-canvas note; escape handled at the plot edge.

Teach me Hide Text

Speed

Guided Inquiry

1 · Meet the energy diagram

Let’s get to know the diagram. The curve is V(x) = s(x³/3 − x). The cyan dashed line is the total energy E, and the amber dashes mark the barrier top V_b at x = −1 m. With E below V_b, the ball is trapped in the right-hand valley — released from rest at x₀ ≈ 1.8 m.

2 · Predict — where is it fastest?

Here’s your first prediction. Once the ball is released in this bound well, where along the track do you think it moves fastest?

3 · Watch the energy budget

Let’s watch the energy budget at work. Press ▶ Play if needed. The trail dots space out widest at the bottom, and v → 0 at the turning points where V(x) = E. Now compare the cyan v-arrow with the magenta a-arrow: at x = +1 m, a = 0 while v peaks.

4 · Predict — raising E past V_b

Now let’s give the ball more energy. The amber line is the barrier height. If you raise E above V_b, what do you think happens to the ball?

5 · Escape, then the separatrix

E is now above V_b, so watch the one-pass escape. Then, for something stranger, try Snap E = V_b (separatrix): the ball creeps toward x = −1 m but never arrives — the gap halves every t_½ = 0.49 s — forever. That critical energy has a name: the separatrix.

6 · Predict — the two flat points

One last puzzle. Remember, slope sets the acceleration, a = −V′(x)/m. Two points here have zero slope: the valley floor at x = +1 m and the barrier top at x = −1 m.

A ball rests at each one, then gets a tiny nudge. How do you think the two respond?

Listen

Controls

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Total energy E E = 0.300 J

−1.35 J+1.5 J

Potential scale

Release point x₀

📈 Speed map v(x)

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📏 Measurements

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KE = E − V(x): —
V(x): —
v = √(2KE/m): —
a = −V′(x)/m: —
t since release: —
turning pts: —
barrier V_b: —

m = 1.00 kg · x in m · sim time = real time (1×)

The landscape and its equilibria

Equilibria where V′ = 0: x = ±1 m. V″(+1 m) = +2s > 0 → stable minimum, V(+1) = −⅔s. V″(−1 m) = −2s < 0 → unstable maximum, barrier V_b = V(−1) = ⅔s (= 0.6667 J at s = 1).

Speed from energy — not from slope

v is set by the gap E − V(x): maximal at the minimum, zero at turning points where V(x) = E. The slope only sets the acceleration a(x) = −V′(x)/m.

The separatrix: time to the top diverges

Linearized near the top (ξ = x + 1) at E = V_b: exponential approach with rate λ = √(2s/m); each halving of the gap costs t_½ = ln2/λ. The ball approaches forever — it never arrives.

The ⚠ counterfactual rule (for contrast)

The tested (wrong) rule: speed ∝ steepness, calibration τ = 1 s. It predicts stalling at BOTH x = ±1 m and no dependence on E — contradicted by the observed maximum speed at x = +1 m.

The Physics

Motion in a potential · what this sim shows, the physics behind it, and every control

At a glance

This sim rolls a ball along a potential-energy curve V(x) and draws its fixed total energy E as a line on the same diagram. Reading that one diagram tells you where the ball can go, how fast it moves there, and whether it stays or escapes.

What you'll learn

How to read the ball's speed off an energy diagram.
What the slope of V(x) does to the ball, and what it does not do.
How the total energy decides whether the ball stays or escapes.
What happens exactly on the dividing line, and how the curve's two flat points differ.

The curve V(x) = s(x³/3 − x), the energy line E, and the barrier height V_b at the crest.

What's on screen

Grey curve: the potential energy V(x) at each position.
Dashed cyan line E: the total energy (drag its ⇕ handle).
Dashed amber line: the barrier height V_b, at the crest x = −1 m.
Ball and trail: dots dropped at equal time steps. White dots: v = 0.
Green KE bar, cyan v-arrow and magenta a-arrow on the ball.
Chips, top left: type of motion, speed v, and a stopwatch t near the barrier.

See itRun the inquiry's opening scene. Watch for: the ball turning back each time it reaches the dashed energy line.

The controls

Press ⌖ on a row to see where that control sits in the sim.

Control	What it changes	Range
Total energy E	The ball's energy, fixed during a run. You can also drag the E line on the plot. Below the valley floor (−⅔s) the sim holds E at the floor.	−1.35 to +1.5 J opens at 0.30 J
E below V_b / E above V_b	Presets: E = 0.45 V_b, or E = 1.25 V_b (at most 1.5 J).	buttons
Snap E = V_b	Sets E exactly equal to the barrier height. Press again to return to your previous E.	on / off
Potential scale s	Stretches the whole curve up and down. The flat points stay at x = ±1 m.	0.4–2 opens at 1
Release point x₀	Where the ball starts. Inside the allowed region it sets off to the left. If the curve at x₀ is above the E line, the start moves to the nearest turning point, at rest. You can also drag the ball.	−0.95 to 2.35 m opens at 2.35 m
📈 Speed map v(x)	Plots the speed against x over the region the ball can reach, with a marker at the ball.	panel
📏 Measurements	Live KE, V, v and a at the ball, time since release, the turning points and V_b.	panel
▶ Play / ⏸ Pause	Runs or freezes the motion.	top bar
Speed	Plays the motion faster or slower on screen. The physics stays the same.	0.25×–4×
↻ Reset	Restarts the mode you are in from its beginning.	top bar

Fixed: mass m = 1 kg, no friction, so E stays constant.

Speed comes from the energy gap

With no friction, E = ½mv² + V(x) stays the same everywhere. So the kinetic energy at any x is the vertical gap from the curve up to the E line:

m = 1 kg in the sim.

Bigger gap, faster ball. The gap is largest where V is lowest, at the valley floor x = +1 m.
Turning points. The ball can only be where V(x) ≤ E. Where the curve meets the line, KE = 0 and the ball turns round.

E s

These sliders move the sim too.

See itRelease the ball from rest at E = 0.30 J. Watch for: v rising to 1.39 m/s as the ball crosses the valley floor, and falling to 0 at each turning point.

The slope sets the acceleration, not the speed

The force on the ball is −V′(x), so the slope of the curve gives the acceleration:

At the valley floor the slope is zero, so a = 0, at the very point where v is largest.
At a turning point v = 0, but the slope is not zero. The force is still there, so the ball turns back instead of stopping.
On the steep stretches the speed changes fastest. Steep means a large acceleration, not a large speed.

See itThe same run, now watching the two arrows. Watch for: the magenta a-arrow shrinking to nothing at the valley floor, just as the cyan v-arrow is longest.

Above the barrier, the ball escapes

The crest at x = −1 m is the highest point between the valley and the open slope to the left. Its height is the barrier, V_b = V(−1) = ⅔s (0.667 J at s = 1).

E below V_b: bound. The line meets the curve on both sides, so there are two turning points. With no friction, the ball swings between them forever.
E above V_b: free. The left turning point is gone. The ball crosses the crest still moving, at √(2(E − V_b)/m), and runs off to the left for good.
The starting point is always in the valley, so E alone decides which of the two happens. The steepness of the curve plays no part.

See itRaise E to 0.833 J, above the barrier. Watch for: the ball slowing over the crest without stopping, then leaving to the left as the top-left chip turns to ESCAPED.

At exactly the barrier, the ball never arrives

Set E = V_b. The ball reaches the crest with zero speed, and close to it the speed is proportional to the distance ξ = x + 1 still to go:

λ = √(2s/m). The gap shrinks exponentially.

The remaining gap halves every t_½ = ln 2 · √(m/2s): 0.49 s at s = 1.
Every halving costs the same time, so the ball gets ever closer but never reaches x = −1 m. The stopwatch keeps counting.
This energy is the separatrix, the boundary between bound and free motion. Just below it, each swing slows near the crest: at s = 1 a full swing takes 4.4 s for small swings, 5.3 s at E = 0.30 J, and longer without limit as E approaches V_b.

See itSnap E onto the barrier height. Watch for: the ball crawling ever slower just right of the crest, while the time keeps running.

Two flat points, two kinds of balance

At x = ±1 m the slope is zero, so a ball at rest there feels no force and stays put. Both points are equilibria. The curvature V″ decides what a small nudge does:

Valley floor, V″ = +2s. Off the floor the force points back toward it, so a nudged ball oscillates. Small swings take 2π√(m/2s): 4.44 s at s = 1.
Crest, V″ = −2s. Off the crest the force points away from it, so any nudge grows. The same rate λ = √(2s/m) sets how fast it runs away.

The sim cannot place a ball at rest on the crest; section 06 shows the crest from the approach side.

See itDrop E to just above the valley floor. Watch for: the ball rocking in a small swing about x = +1 m, always pulled back.

Where it matters

A pendulum has the same structure. Below 2mgℓ it swings; above it, it goes over the top. At exactly 2mgℓ it creeps toward upside down and never gets there.
A chemical bond is a potential well. The H₂ molecule needs about 4.5 eV to break: with less, the two atoms vibrate; with more, they fly apart.
Escape speed. A body launched from Earth's surface at 11.2 km/s or more (ignoring air) has enough energy to climb out of Earth's gravity well and never return.

Go deeper: the pendulum's energy diagram

V(θ) = mgℓ(1 − cos θ). The crests at 2mgℓ are the upside-down position; the amber line is the pendulum's separatrix.

Summary

In one line. The height of E above the curve sets the speed; the height of E against the barrier sets the fate.

Key points

KE = E − V(x): speed peaks where V is lowest and is zero at the turning points.
The slope sets the acceleration, a = −V′(x)/m, not the speed.
Below V_b the ball is bound and swings forever; above V_b it escapes once.
At E = V_b exactly, the ball approaches the crest forever: the separatrix.

Common mix-ups

"Steeper means faster." Steep means the speed is changing fast. The speed is largest where the slope is zero, at the valley floor.
"Energy is conserved, so the speed is the same everywhere." E is constant, but its split between KE and V changes at every x.
"At E = V_b the ball reaches the top and stops there." It gets ever closer but never arrives.

Things to try on your own

Set s = 2, then raise E until the ball escapes. Check that the threshold matches V_b in Measurements (1.333 J).
Keep E fixed and move x₀ across the valley. Check that the turning points and top speed do not change.