Conservation Laws: Energy & Momentum

Name: ________________________________ Section: __________ Date: __________

Station: __________ Group members: ________________________________________________

Goal: Use the demo to make a claim supported by (1) at least one number/readout and (2) at least one sanity check.

Station card: Conservation Laws (Orbits) (8–10 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. Turning points: On Elliptical, predict first: where on the orbit is the body neither approaching nor receding from the Sun? Then switch the stage to Potential and press Step until the gold line has zero length. Is the body at the $r_p$ label or the $r_a$ label? Find the other place the line vanishes.
3. Direction check: Change direction to $60^\circ$. Does the escape speed factor change?
4. What does change: At a fixed speed factor, compare $h$ and periapsis $r_p$ at $0^\circ$ vs $60^\circ$.
5. Explanation (1–2 sentences): Use “energy sets bound vs unbound”, “angular momentum sets closest approach” and “the orbit turns around where $U_{\rm eff}(r) = \varepsilon$.”

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.
• Effective potential $U_{\rm eff}(r)$ (Potential view): the energy landscape. The body can only be where the energy line $\varepsilon$ is above it; the gold line between them is $\tfrac{1}{2}v_r^2$.

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 $\varepsilon$.