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

Your station artifact (fill in):

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

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 $\varepsilon$.