Conservation Laws: Energy & Momentum

Overview

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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)

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.”)
Decrease speed: set $v/v_{\rm circ}\approx 0.75$. Ask: “Does it still stay at the same radius?” (No — it becomes elliptical.)
Go to escape: set $v/v_{\rm circ}=\sqrt{2}\approx 1.414$. Ask: “What changes qualitatively?” (It no longer returns; it’s the escape boundary.)
Go beyond escape: set $v/v_{\rm circ}\approx 1.8$. Ask: “What should the orbit do now?” (Hyperbolic flyby.)
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.

Activities

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

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

Set speed factor to 1.00 (circular) and ask: “Bound or unbound?”
Set speed factor to 1.30 and ask: “Still bound?”
Set speed factor to 1.414 (escape) and ask: “What’s special about this value?”

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

MW Short (8–12 min): Why √2?

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

Keep $M$ and $r_0$ fixed. Record $v_{\rm circ}$ by setting speed factor = 1.
Increase the speed factor until the orbit switches to “parabolic (escape).”
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):

Escape test: Find the speed factor where the orbit becomes “escape/parabolic” (about $\sqrt{2}$).

Direction check: Change direction to $60^\circ$. Does the escape speed factor change?

What does change: At a fixed speed factor, compare $h$ and periapsis $r_p$ at $0^\circ$ vs $60^\circ$.

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 ($r_p$).

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 $\approx 1.414$), 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 $\varepsilon$.

Assessment

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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. 1.41
D. 2.00

Answer: C
Why: escape occurs at $v_{\rm esc}=\sqrt{2},v_{\rm circ}\approx 1.414,v_{\rm circ}$.

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: $h=|\mathbf{r}\times\mathbf{v}|=rv\sin\phi$ is smallest when motion is nearly radial (small sideways component).

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 √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:

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

Ideal answers:

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

Model notes (deeper)

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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:

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

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.

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:

$$h = |\mathbf{r}\times\mathbf{v}|$$

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 length^2/time

✓ Units match.

Circular vs escape speed (the most teachable relationship)

At a given radius $r$:

$$v_{\rm circ} = \sqrt{\frac{\mu}{r}}$$

$$v_{\rm esc} = \sqrt{\frac{2\mu}{r}}$$

So:

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

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:

$$r(\nu) = \frac{p}{1 + e\cos\nu}$$

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:

$$p = \frac{h^2}{\mu}$$

and the eccentricity vector:

$$\mathbf{e} = \frac{\mathbf{v}\times\mathbf{h}}{\mu} - \hat{\mathbf{r}}$$

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:

$$G = 4\pi^2\ \frac{\mathrm{AU}^3}{\mathrm{yr}^2,M_\odot}$$

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

$\varepsilon$ is in AU^2/yr^2
$h$ is in AU^2/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.

Backlog

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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

Priority	Impact	Effort	Category	Notes	Code entrypoint
P0	High	Medium	Feature	DONE (2026-01-29): Add a standalone “orbit shapes from conservation laws” demo (analytic two-body; classifies bound/unbound).	`apps/demos/src/demos/conservation-laws/`
P1	High	Medium	Docs	DONE (2026-01-29): Create instructor resources (index, model, activities, assessment, backlog).	`demos/_instructor/conservation-laws/`
P1	High	Medium	Physics	DONE (2026-01-29): Centralize mechanics time/length conventions in shared `AstroConstants` and use shared two-body analytic helpers.	`demos/_assets/physics/`

Active backlog

Priority	Impact	Effort	Category	Notes	Code entrypoint
P1	High	Medium	Pedagogy	Add an “energy decomposition” toggle that explicitly shows $v^2/2$, $-\mu/r$, and $\varepsilon$ (and makes the “sign of $\varepsilon$” story unavoidable).	`apps/demos/src/demos/conservation-laws/main.ts`
P1	High	Medium	Pedagogy	Add an “equal areas” overlay (wedge + constant areal velocity readout) to connect directly to Kepler’s 2nd law.	`apps/demos/src/demos/conservation-laws/`
P1	Medium	Medium	UX	Add an option to choose the initial position angle (currently fixed at +x), so students can test invariance under rotation.	`packages/physics/src/conservationLawsModel.ts` + `apps/demos/src/demos/conservation-laws/main.ts`
P2	Medium	Low	UX	Add a unit toggle (AU/yr <-> km/s <-> CGS) for $\varepsilon$ and $h$ readouts (keeps units consistent across the “mechanics suite”).	`apps/demos/src/demos/conservation-laws/main.ts`
P2	Medium	Medium	Pedagogy	Add a “station mode” overlay: numbered prompts, prediction checkpoints, and a small data table students can copy/paste.	`apps/demos/src/demos/conservation-laws/`
P3	Medium	High	Physics	Add an optional “integrator preview” mode (Euler vs symplectic vs RK4) that shows conservation drift — defer until the numerical-integrators project.	`apps/demos/src/demos/conservation-laws/` + `packages/physics/src/*`

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)