Instructor notes: Conservation Laws: Energy & Momentum

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This guide is instructor-facing Student demo: /play/conservation-laws/
Main code: apps/demos/src/demos/conservation-laws/main.ts
Shared physics: packages/physics/src/twoBodyAnalytic.ts
Plot helpers: packages/physics/src/conservationLawsModel.ts

Where to go next

• Model + math + assumptions: model.md
• In-class activities (MW quick + Friday lab + station version): activities.md
• Assessment bank (clickers + short answer + exit ticket): assessment.md
• Future enhancements (planning backlog): backlog.md

Why this demo exists

Why This Matters Students often learn “ellipses, parabolas, and hyperbolas” as disconnected shapes. In orbital mechanics, those shapes are not arbitrary: they are determined by conservation laws. This demo makes a single big idea concrete:

If you know the conserved specific energy $\varepsilon$ and specific angular momentum $h$, you know the orbit type.

That’s a durable mental model that transfers to escape velocity, bound vs unbound systems, and later to numerical integration (where conservation drift becomes a diagnostic).

Learning goals

ASTR 101

Students should be able to:

• Predict whether an object is bound (returns) or unbound (escapes) based on speed
• Explain why escape speed is larger than circular speed
• Describe how “more sideways motion” means more angular momentum and therefore a “less radial” trajectory

ASTR 201 / Mechanics

Students should also be able to:

• Use $\varepsilon = v^2/2 - \mu/r$ to classify bound vs unbound motion
• Use $v_{\rm circ}=\sqrt{\mu/r}$ and $v_{\rm esc}=\sqrt{2\mu/r}$ and explain why $v_{\rm esc}=\sqrt{2}\,v_{\rm circ}$
• Interpret $h = |\mathbf{r}\times\mathbf{v}|$ as the control knob for periapsis distance and areal sweep rate

10–15 minute live-teach script (projector)

1. Start at the default: $M=1\,M_\odot$, $r_0=1\,\mathrm{AU}$, speed factor $v/v_{\rm circ}=1$, direction $0^\circ$. Ask: “What do you predict the orbit looks like?” (Most students say “circle.”)

2. Decrease speed: set $v/v_{\rm circ}\approx 0.75$. Ask: “Does it still stay at the same radius?” (No — it becomes elliptical.) Press Play and ask: “Which readouts change, and which stays fixed?” ($K$ and $U$ change; $\varepsilon$ does not.)

3. Go to escape: press the Escape preset, which sets $v/v_{\rm circ}=\sqrt{2}\approx 1.414$ exactly (the slider alone steps from 1.41, still bound, to 1.42, unbound).

4. Go beyond escape: set $v/v_{\rm circ}\approx 1.8$. Ask: “What should the orbit do now?” (Hyperbolic flyby.)

5. Change direction: increase the direction magnitude (e.g. $60^\circ$). Ask: “We kept the speed factor similar — why did the closest approach change?” Connect: changing direction changes $h$.

Suggested connections to other demos

• Kepler’s Laws: This demo explains why Keplerian orbits take conic shapes in the first place.
• Binary Orbits: The relative orbit is set by the same conservation laws (now with $M_1+M_2$).
• (Future) Numerical Integrators: Conservation drift becomes a visual test of algorithm quality.

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MW Quick (3–5 min): Bound vs unbound (prediction first)

Setup (projector): Default settings ($M=1\,M_\odot$, $r_0=1\,\mathrm{AU}$).

1. Set speed factor to 1.00 (circular) and ask: “Bound or unbound?”
2. Set speed factor to 1.30 and ask: “Still bound?”
3. Press the Escape preset (speed factor exactly $\sqrt{2}\approx 1.414$) and ask: “What’s special about this value?”

Key takeaway: the sign of $\varepsilon$ changes at escape.

MW Short (8–12 min): Why $\sqrt{2}$?

Goal: students discover $v_{\rm esc}=\sqrt{2}\,v_{\rm circ}$ using the demo’s readouts.

1. Keep $M$ and $r_0$ fixed. Record $v_{\rm circ}$ by setting speed factor = 1.
2. Increase the speed factor until the orbit type switches from elliptical (1.41) to hyperbolic (1.42), then press Escape to see the exact boundary, “parabolic (escape)”.
3. Compute the ratio $v_{\rm esc}/v_{\rm circ}$ from the speed factor and compare to $\sqrt{2}$.

Discussion prompt: “Why does energy care about speed squared?” Tie back to $\varepsilon=v^2/2-\mu/r$.

Friday Lab (20–30+ min): Map orbit type in (speed, direction) space

Part A: Build a classification map

Students collect a small dataset by varying:

• speed factor $v/v_{\rm circ}$
• direction angle ($0^\circ$ tangential; near $\pm 85^\circ$ radial)

Deliverable: a table with columns:

• speed factor
• direction angle
• orbit type (circular / elliptical / parabolic / hyperbolic)
• $e$
• $\varepsilon$
• $h$

Part B: Claim–Evidence–Reasoning

Claim: “Orbit type depends primarily on energy, while closest approach depends strongly on angular momentum.”

Evidence: use at least two paired comparisons where speed factor is similar but direction differs, producing noticeably different $h$ and periapsis distance.

Reasoning: connect to:

• $\varepsilon = v^2/2 - \mu/r$ (bound vs unbound)
• $h = |\mathbf{r}\times\mathbf{v}|$ (controls periapsis via $p=h^2/\mu$)

Station version (6–8 min)

Station card: Conservation Laws (Orbits) (6–8 minutes) Setup: Use $M=1\,M_\odot$ and $r_0=1\,\mathrm{AU}$ (defaults).

Your station artifact (fill in):

1. Escape test: Raise the speed factor until the orbit type changes from elliptical to hyperbolic. Record the last elliptical and first hyperbolic values, then press Escape and record the exact value it sets.
2. Direction check: Change direction to $60^\circ$. Does the escape speed factor change?
3. What does change: At a fixed speed factor, compare $h$ and periapsis $r_p$ at $0^\circ$ vs $60^\circ$.
4. Explanation (1–2 sentences): Use “energy sets bound vs unbound” and “angular momentum sets closest approach.”

Word bank + sanity checks Word bank:

• Speed factor ($v/v_{\mathrm{circ}}$): speed compared to circular speed at the same $r_0$.
• Specific energy $\varepsilon$: determines bound ($\varepsilon<0$) vs escape ($\varepsilon=0$) vs hyperbolic ($\varepsilon>0$).
• Angular momentum $h$: depends on the tangential part of the velocity; it controls how close the orbit swings in ($r_p$).
• Kinetic $K$ and potential $U$: they trade places as the body moves; their sum $\varepsilon$ does not change.

Key relationship (specific orbital energy):

$$\varepsilon=\frac{v^2}{2}-\frac{\mu}{r}$$

Sanity checks:

• Escape happens at:

  $$v_{\mathrm{esc}}=\sqrt{2}\,v_{\mathrm{circ}}$$

  (so speed factor $\approx 1.414$; the slider steps from 1.41 to 1.42), regardless of direction.

• Changing direction changes $h$ (and therefore $r_p$), even if the speed magnitude stays the same.

• “Bound vs unbound” tracks the sign of $\varepsilon$.

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• Student demo: /play/conservation-laws/
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Clicker questions (ASTR 101)

Q1: Escape threshold (conceptual)

Using the demo at $M=1\,M_\odot$ and $r_0=1\,\mathrm{AU}$, you slowly increase speed factor $v/v_{\rm circ}$. At what value does the orbit stop being bound?

A. 1.00
B. 1.20
C. $\sqrt{2}\approx 1.414$
D. 2.00

Answer: C
Why: escape occurs at $v_{\rm esc}=\sqrt{2}\,v_{\rm circ}$. On the slider the orbit is still elliptical at 1.41 and already hyperbolic at 1.42; the Escape preset sets the exact value.

Q2: Direction and angular momentum

At the same radius and same speed factor, which direction produces the smallest angular momentum magnitude?

A. $0^\circ$ (purely tangential)
B. $30^\circ$
C. $60^\circ$
D. $85^\circ$ (near radial)

Answer: D
Why: $h = r v\cos\theta$, where $\theta$ is the direction from tangential, so only the sideways part of the velocity counts; at $85^\circ$ that is $\cos 85^\circ \approx 0.09$ of the speed.

Short-answer (ASTR 201)

SA1: Energy classification

Write down the specific energy equation and explain how its sign classifies orbit type.

Expected elements:

• $\varepsilon = v^2/2 - \mu/r$
• $\varepsilon<0$ bound, $\varepsilon=0$ escape, $\varepsilon>0$ unbound

SA2: Why $\sqrt{2}$?

Derive $v_{\rm esc}=\sqrt{2}\,v_{\rm circ}$ at fixed $r$.

Expected elements:

• circular orbit: set centripetal requirement or use energy with $a=r$
• escape: set $\varepsilon=0$
• show ratio $\sqrt{2}$

Exit ticket (2 minutes)

One sentence each:

1. “Energy tells you ___.”
2. “Angular momentum tells you ___.”

Ideal answers:

1. Energy tells you whether the orbit is bound or unbound.
2. Angular momentum tells you how close the orbit can approach (and the areal sweep rate).

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What the demo computes (in one sentence)

Given an initial position $\mathbf{r}_0$ and velocity $\mathbf{v}_0$ around a central mass, the demo computes the conserved quantities $\varepsilon$ and $h$ and uses them to infer the orbit type (bound vs escape vs flyby).

The two conserved quantities

1) Specific orbital energy

The specific (per unit mass) orbital energy is:

Let’s unpack each piece:

• $\varepsilon$ is specific energy (energy per unit mass)
• $v$ is the speed (same distance/time units as the demo state)
• $r$ is the distance from the central mass
• $\mu$ is the gravitational parameter: $\mu = GM$

What this equation is really saying: kinetic energy per mass ($v^2/2$) competes with gravitational potential per mass ($-\mu/r$). Their sum stays constant in a two‑body Newtonian system.

The demo shows the two pieces as separate readouts:

During Play, $K$ and $U$ change as the body moves while $\varepsilon$ stays fixed: the conservation law made visible.

Orbit classification from $\varepsilon$:

• If $\varepsilon < 0$, the motion is bound (ellipse; includes the circular case).
• If $\varepsilon = 0$, the motion is exactly at escape (parabola).
• If $\varepsilon > 0$, the motion is unbound (hyperbola).

2) Specific angular momentum

The specific angular momentum is:

Let’s unpack each piece:

• $h$ is specific angular momentum (per unit mass)
• $\mathbf{r}$ is the position vector
• $\mathbf{v}$ is the velocity vector

What this equation is really saying: “sideways motion at large radius” produces large angular momentum. Large $h$ prevents deep plunges; small $h$ allows close approaches.

Dimensional check:

• $\mathbf{r}$ has units of length
• $\mathbf{v}$ has units of length/time
• so $h$ has units of $\mathrm{length}^2/\mathrm{time}$

✓ Units match.

Circular vs escape speed (the most teachable relationship)

At a given radius $r$:

So:

What this is really saying: escape speed is only about 41% larger than circular speed at the same radius — a powerful intuition for why “a little extra speed” can unbind an orbit.

How the demo draws the orbit (conic geometry)

The orbit is plotted using the conic-section polar form:

Let’s unpack each piece:

• $\nu$ is the true anomaly (angle from periapsis)
• $e$ is eccentricity (shape parameter)
• $p$ is the semi‑latus rectum

The demo computes:

and the eccentricity vector:

with $e = |\mathbf{e}|$. The direction of $\mathbf{e}$ points toward periapsis and sets the orbit’s orientation in the plot.

Units used in this demo

The demo’s UI uses:

• distance in AU
• time in years

Internally, it uses the teaching normalization:

So $\mu = GM$ is in $\mathrm{AU}^3/\mathrm{yr}^2$, and:

• $\varepsilon$ is in $\mathrm{AU}^2/\mathrm{yr}^2$
• $h$ is in $\mathrm{AU}^2/\mathrm{yr}$

What’s simplified / not modeled

• Motion is planar (2D).
• No perturbations (no other planets/stars).
• No relativity.
• The orbit path is drawn from conic geometry; the demo does not attempt ephemeris-grade timing along hyperbolic trajectories.

Navigation

• Instructor hub: All instructor notes
• Back to this demo guide: Guide
• Student demo: /play/conservation-laws/
• This demo: Model · Activities · Assessment · Backlog

How to use this backlog This is a planning guide. Prefer changes that increase correctness and reduce cognitive friction before adding new features.

Completed items

Active backlog

Priority definitions

• P0: Correctness or critical functionality (must fix before use)
• P1: High-impact pedagogy or usability (should add soon)
• P2: Nice-to-have enhancements (add when time permits)
• P3: Future extensions (research-level or specialized topics)