Chapter 2 — Key Takeaways (Newton's Laws in Space)

A one-page reference. Reread this before an exam, or before you compute your first orbit.

Newton's three laws, translated to spaceflight

Law Classroom form What it means in space
First (inertia) Motion persists unless a net force acts. A coasting spacecraft needs no fuel to keep moving; fuel only changes velocity. An orbit is a perpetual free fall and does not decay in vacuum.
Second $\mathbf{F} = m\mathbf{a}$ (constant mass) The true form is $\mathbf{F} = d\mathbf{p}/dt$. A rocket's mass changes, so use momentum. Thrust $= \dot m_p\,v_e$.
Third Equal and opposite reactions. The only vacuum propulsion is to throw mass backward and ride the reaction. A rocket pushes on its own exhaust, not the air.

Key equations (with symbols and units)

Equation Meaning Symbols / units
$\mathbf{p} = m\mathbf{v}$ momentum $m$ (kg), $\mathbf{v}$ (m/s) → $\text{kg}\cdot\text{m/s}$
$\dfrac{d\mathbf{p}_{\text{total}}}{dt} = 0$ conservation of momentum (no external force) total $m\mathbf{v}$ is constant
$T = \dot m_p\,v_e$ rocket thrust $\dot m_p$ (kg/s), $v_e$ (m/s) → newtons
$F = G\dfrac{m_1 m_2}{r^2}$ universal gravitation $G = 6.674\times10^{-11}\ \text{N}\cdot\text{m}^2/\text{kg}^2$
$\mu = GM$ gravitational parameter Earth: $3.986\times10^{5}\ \text{km}^3/\text{s}^2$
$a = \dfrac{\mu}{r^2}$ gravitational acceleration independent of the falling object's mass
$U = -\dfrac{\mu m}{r}$ gravitational potential energy zero at infinity; negative in the "well" (J)
$v_{\text{esc}} = \sqrt{\dfrac{2\mu}{r}}$ escape velocity Earth surface: $\approx 11.2\ \text{km/s}$
$v_{\text{esc}} = \sqrt{2}\,v_{\text{circ}}$ escape vs. circular orbit escape is $\approx 41\%$ faster than orbit

The two great speeds (at Earth's surface radius)

Speed Formula Value
Circular orbit $v_{\text{circ}} = \sqrt{\mu/R}$ $\approx 7.9\ \text{km/s}$
Escape $v_{\text{esc}} = \sqrt{2\mu/R}$ $\approx 11.2\ \text{km/s}$

Ratio is always $\sqrt{2}$. Once in orbit you already carry most of the energy to escape — the origin of "in orbit, you're halfway to anywhere."

Body data (memorize the order of magnitude)

Body $\mu$ (km³/s²) Radius (km) Surface $g$ (m/s²) $v_{\text{esc}}$ (km/s)
Earth $3.986\times10^{5}$ $6{,}371$ $9.82$ $11.2$
Moon $4.903\times10^{3}$ $1{,}737$ $1.62$ $2.38$
Mars $4.283\times10^{4}$ $3{,}390$ $3.73$ $5.03$
Sun $1.327\times10^{11}$ $696{,}000$ $274$ $618$

Decision aid — "which idea do I use?"

You want to explain / find… Use
why a satellite stays up with no fuel first law + gravity (perpetual free fall)
the thrust of an engine $T = \dot m_p v_e$
the recoil / speed gained by throwing mass conservation of momentum, $m_1 v_1 = m_2 v_2$
gravity at some altitude $a = \mu/r^2$ (inverse square)
the speed to leave a body forever $v_{\text{esc}} = \sqrt{2\mu/r}$
the energy to lift 1 kg out of a well $\mu/R$ (J/kg) $= \tfrac12 v_{\text{esc}}^2$

Common pitfalls

Pitfall Reality
"A rocket pushes against the air/space." It reacts against its own exhaust; it works better in vacuum.
"There's no gravity in space." Gravity at the ISS is ~$88\%$ of surface. Floating is free fall, not zero gravity.
"Escape velocity is a speed you must maintain." It's a ballistic (coasting) threshold; a powered craft can leave slowly.
"Escape velocity depends on the ship's mass." It doesn't — $m$ cancels ($v_{\text{esc}} = \sqrt{2\mu/r}$).
"$F = ma$ handles a rocket." Use $F = d\mathbf{p}/dt$; a rocket's mass changes. The thrust term uses $v_e$, not the rocket's ground speed.
"Orbits never decay." True only in a perfect vacuum; real low orbits decay from residual drag (station-keeping, Ch. 12).

Numbers worth memorizing

  • Earth $\mu_\oplus = \mathbf{3.986\times10^{5}\ \text{km}^3/\text{s}^2}$; $G = 6.674\times10^{-11}\ \text{N}\cdot\text{m}^2/\text{kg}^2$.
  • Earth escape velocity $\approx \mathbf{11.2\ \text{km/s}}$; surface orbital speed $\approx 7.9\ \text{km/s}$.
  • $v_{\text{esc}}/v_{\text{circ}} = \sqrt{2} \approx 1.41$.
  • Earth's gravity well $\approx \mathbf{63\ \text{MJ/kg}}$ deep ($= \mu_\oplus/R_\oplus = \tfrac12 v_{\text{esc}}^2$).
  • Gravity at the ISS ($\sim 420\ \text{km}$) $\approx 8.6\ \text{m/s}^2$ ($\approx 88\%$ of surface).

Mission / astrotools additions this chapter

  • MDR: recorded your mission's Primary Body data block — $\mu$, radius, surface $g$, escape velocity.
  • Code: previewed escape_velocity(mu, r) = sqrt(2*mu/r), which joins orbits.py in Chapter 6.

The one-sentence bridge to Chapter 3

Conservation of momentum (§2.4), applied not to one puff of gas but to a rocket burning continuously and growing lighter, integrates into the Tsiolkovsky rocket equation $\Delta v = v_e \ln(m_0/m_f)$ — the subject of Chapter 3, and the master equation of the whole book.