Newton’s Revolution

From Patterns to Physics

Dr. Anna Rosen

February 4, 2026

The Apple and the Moon

Does the force that pulls an apple to the ground…

also reach up to the Moon?

0–1 min. “Whether or not the story happened exactly this way, it captures Newton’s core insight.”

The Moon Is Falling

The Moon IS falling toward Earth.

Right now. Continuously. Every second.

But it’s moving sideways so fast that it keeps missing.

That combination of falling + moving sideways is what we call an orbit.

1–2 min. “Before Newton, ‘earthly’ physics and ‘celestial’ physics were separate. Newton unified them with one equation.”

The Moon’s Orbit: Velocity + Inward Force

Two ideas at once:

• Sideways velocity carries the Moon forward
• Gravity pulls inward

The path curves because both happen continuously.

Credit: cococubed.com

Use the force/velocity arrows to explain why the Moon keeps missing Earth.

Newton’s Cannon

Credit: cococubed.com

Fire fast enough to fall around Earth instead of hitting it.

2–3 min. “Orbiting is falling, but moving sideways fast enough to keep missing the ground.”

Today’s Learning Objectives

By the end of today, you can:

• State Newton’s three laws and use \(F = ma\)
• Explain why circular motion needs centripetal acceleration
• State universal gravitation and the inverse-square rule

Today’s Learning Objectives (continued)

By the end of today, you can:

• Distinguish mass from weight
• Explain why astronauts float (free fall, not zero gravity)
• Connect Newton’s gravity to Kepler’s three laws
• Use orbital motion to infer mass of a central object

3–4 min. “This is the lecture that explains why Kepler’s patterns exist.”

Where We Left Off (Lecture 5)

Kepler discovered three empirical laws:

1. Orbits are ellipses with the Sun at one focus
2. Planets sweep equal areas in equal times
3. \(P^2 \propto a^3\) (period-distance relation)

These describe what planets do.

But Kepler couldn’t explain why.

4–5 min. “Today, Newton gives us the why.”

Part 1: The Rules of Motion

Newton’s Three Laws

Newton’s First Law: Inertia

An object at rest stays at rest. An object in motion stays in motion — unless acted upon by a net force.

Key insight: Things don’t change their motion on their own.

If something speeds up, slows down, or changes direction — a net force caused it.

5–6 min. “This contradicts Aristotle, who thought motion requires continuous effort.”

First Law Example: Deep Space

A spacecraft in deep space fires its engines briefly, then shuts them off. What happens next?

• It gradually slows down and stops
• It continues at constant velocity indefinitely
• It starts drifting toward the nearest star
• It needs to keep firing to maintain speed

1 min. iClicker poll. “No friction in space = no slowing down.”

Newton’s Second Law: F = ma

\[\vec{F}_{net} = m\vec{a}\]

Net force tells matter how to accelerate.

Mass tells matter how much to resist.

A vector has magnitude and direction. The arrows show direction.

Units: force in N, mass in kg, acceleration in \(\mathrm{m/s^2}\).

More net force, more acceleration.

More mass, less acceleration.

6–7 min. “The arrow notation means these are vectors — direction matters.”

Quick Check: F = ma

The same force is applied to a bowling ball and a tennis ball. Which experiences greater acceleration?

• The bowling ball (more mass)
• The tennis ball (less mass)
• Both experience the same acceleration
• Neither accelerates

1 min. iClicker poll.

Critical Insight: Direction Changes = Acceleration

Constant speed ≠ constant velocity

Acceleration happens when either changes:

• Speeding up
• Slowing down
• Changing direction at constant speed

A planet moving in a circle at constant speed is accelerating because its direction constantly changes!

7–8 min. “This is the key to understanding orbits. Circular motion requires a force.”

Quick Check: Circular Motion

A car drives around a circular track at a constant speedometer reading of 60 mph. Is the car accelerating?

• No — the speed is constant
• Yes — the direction is constantly changing
• Only if the driver presses the gas pedal
• Only if there’s friction

1 min. iClicker poll. “The tires push sideways — that’s the centripetal force.”

Centripetal Acceleration

For circular motion at speed \(v\) and radius \(r\):

\[a_c = \frac{v^2}{r}\]

This acceleration points toward the center.

The force required:

\[F_c = ma_c = \frac{mv^2}{r}\]

8–9 min. “Centripetal means ‘center-seeking.’ Something has to provide this force.”

What Provides the Centripetal Force?

Centripetal force is not a new type of force.

It’s whatever force happens to be pulling the object toward the center:

• For planets: gravity
• For a ball on a string: tension
• For a car turning: friction between tires and road

9–10 min. “For orbits, gravity provides the centripetal force.”

Circular Motion Needs an Inward Force

If the inward force vanishes, the object keeps moving straight.

That is Newton’s First Law in action.

Credit: cococubed.com

Shows why inward force is required for circular motion.

Quick Check: Centripetal Force

A planet orbits the Sun in a nearly circular orbit. What force provides the centripetal acceleration?

• The planet’s inertia
• The planet’s momentum
• Gravity from the Sun
• The planet’s rotational spin

1 min. iClicker poll.

Newton’s Third Law: Action-Reaction

For every force, there is an equal and opposite reaction force.

Examples:

• You push on the ground; the ground pushes back on you
• Earth pulls on the Moon; the Moon pulls on Earth
• The Sun pulls on Jupiter; Jupiter pulls on the Sun

10–11 min. “The forces are equal. The accelerations are not — because masses differ.”

The Astronomical Payoff

Jupiter pulls on the Sun with the same force the Sun pulls on Jupiter.

But the Sun is about 1000 times more massive, so it accelerates about 1000 times less.

The Sun wobbles — just barely.

This tiny stellar wobble is one major way we detect exoplanets!

11–12 min. “The radial velocity method — we see the star wobble and infer the planet.”

Center of Mass: The Real Pivot Point

The center of mass is the balance point of a system. If you could put your finger exactly there, the system would not tip.

For a star–planet pair, both objects orbit this point, not each other. The more massive object sits closer to it, so its orbit is smaller — that is the wobble we detect.

Credit: cococubed.com

Tie directly to stellar wobble and exoplanet detection.

Quick Check: Third Law

Earth exerts a gravitational force on the Moon. The Moon exerts a force on Earth that is:

• Much weaker, because the Moon is smaller
• Much stronger, because Earth is closer
• Exactly equal in magnitude, opposite in direction
• Zero, because the Moon is in space

1 min. iClicker poll. “Forces equal, accelerations differ.”

Part 2: Universal Gravitation

One equation rules all

Predict First!

If you double your distance from Earth’s center, gravity becomes…

• 1/2 as strong
• 1/4 as strong
• 1/8 as strong

30 sec. Have students vote before revealing. Reveal: 1/4 (inverse-square).

Newton’s Law of Universal Gravitation

\[F_g = \frac{Gm_1m_2}{r^2}\]

Part	Meaning
\(G\)	Universal constant (\(6.67 \times 10^{-11}\ \mathrm{N\,m^2/kg^2}\))
\(m_1 m_2\)	Force depends on both masses
\(r^2\)	Force weakens with distance squared

12–14 min. “The same equation works for apples, moons, and galaxies.” Assumptions (briefly): works well when objects act like point masses / spherically symmetric bodies, and in the Newtonian regime.

The Gravity Equation in Pictures

Gravity depends on:

• Both masses (\(m_1, m_2\))
• Distance squared (\(r^2\))

Closer or more massive means stronger force.

Credit: cococubed.com

14–15 min. Tie each symbol to a part of the image.

The Inverse-Square Law

“\(r^2\) in the denominator” is profound:

Distance change	Force change
Double the distance	Force drops to 1/4
Triple the distance	Force drops to 1/9
\(10\times\) the distance	Force drops to 1/100

15–16 min. “Like light spreading from a flashlight — same total, more area, less per spot.” Physical intuition: sphere area grows like \(r^2\), so effect per area drops like \(1/r^2\).

Why Inverse-Square?

As distance grows, the same influence spreads over a larger sphere.

Area grows like \(r^2\), so strength drops like \(1/r^2\).

What to notice: inverse-square comes from spreading over spherical area — double distance means 4× the area and 1/4 the per-area strength. (Credit: Illustration: A. Rosen (SVG))

16–17 min. Visual explanation of inverse-square spreading.

Quick Check: Inverse-Square

The Moon is about 60 Earth radii from Earth’s center. The Moon experiences gravitational acceleration that is:

• 60 times weaker
• 360 times weaker
• 3,600 times weaker
• The same strength

1–2 min. iClicker poll, then explain. Reasoning: at \(60\times\) the distance, \(1/60^2 = 1/3600\).

Mass vs. Weight

A critical distinction

The Distinction

Quantity	What it measures	Changes with location?
Mass (\(m\))	Resistance to acceleration (inertia)	No — same everywhere
Gravitational force (\(F_g\))	Force from gravity	Yes — depends on local \(g\)
Apparent weight	What you feel (scale reading)	Yes — depends on gravity AND motion

17–18 min. “Mass is intrinsic. Weight depends on where you are.”

Quick Check: Mass vs. Weight

An astronaut has mass 70 kg on Earth. On the Moon (about \(1/6\) of Earth’s \(g\)), their:

• Mass decreases to about 12 kg
• Weight is about 1/6, but mass stays 70 kg
• Both mass and scale reading stay the same
• Mass increases because there’s less gravity

1 min. iClicker poll.

Why Astronauts Float

Fixing the biggest misconception

The Wrong Answer

Misconception: “Astronauts float because there’s no gravity in space.”

At ISS altitude (~400 km), Earth’s gravity is still ~90% as strong as at the surface!

18–19 min. “This is one of the most common misconceptions in physics.”

The Right Answer

Astronauts float because they’re in free fall.

The ISS and everyone inside are falling toward Earth together.

But moving sideways so fast they keep missing.

Everything falls together at the same rate, so nothing pushes back on you.

That means no sensation of weight.

19–20 min. “You haven’t escaped gravity. You’ve eliminated the floor’s push.”

Free Fall Feels Weightless

You feel weight when a surface pushes back on you.

In orbit, everything falls together, so there is no push.

What to notice: weight is the floor’s push (normal force) — in free fall there is gravity but no supporting push, so you feel weightless. (Credit: Illustration: A. Rosen (SVG))

Free fall = apparent weightlessness

Not because gravity is gone — because everything is falling together.

20–21 min. Let the image drive it home.

Quick Check: Weightlessness

An astronaut floats inside the ISS, 400 km above Earth. Earth’s gravitational pull on the astronaut is:

• Zero — that’s why they float
• About 90% of what it would be on Earth’s surface
• Exactly the same as on Earth’s surface
• Much stronger than on Earth’s surface

1 min. iClicker poll.

Part 3: Newton Explains Kepler

From patterns to physics

The Big Picture

Kepler’s Law	Newton’s Explanation
Orbits are ellipses	Inverse-square force implies conic sections
Equal areas in equal times	Central force implies angular momentum conservation
\(P^2 \propto a^3\)	Inverse-square gives this exact exponent

Kepler said “here’s what happens.”

Newton said “here’s why it must happen.”

21–23 min. “All three laws flow from one equation.”

Explaining Kepler I: Ellipses

Question: Why are orbits ellipses?

Newton’s Answer:

Any object moving under an inverse-square central force traces a conic section.

• Not enough energy to escape: ellipse
• Just enough energy to escape: parabola
• More than escape energy: hyperbola

23–24 min. “The shape is a CONSEQUENCE of inverse-square, not an assumption.”

Orbits Are Conic Sections

Energy decides the shape:

• Ellipse: bound
• Parabola: just escapes
• Hyperbola: unbound

Credit: cococubed.com

24–25 min. Visual of conic sections.

Quick Check: First Law Explained

Newton explained that planetary orbits are ellipses because:

• Planets were created in elliptical paths
• Inverse-square gravity mathematically produces conic sections
• Ellipses are the simplest possible curves
• The Sun’s rotation shapes the orbits

1 min. iClicker poll.

Explaining Kepler II: Equal Areas

Question: Why do planets sweep equal areas in equal times?

Newton’s Answer:

Gravity is a central force — always pointing toward the Sun.

Central forces conserve angular momentum.

If \(L\) is constant and you get closer (\(r\) smaller), you must move faster (\(v\) larger).

25–26 min. “Same physics as the ice skater pulling arms in.”

Equal Areas = Variable Speed

In the same time interval:

• Near perihelion: fast, short arc
• Near aphelion: slow, long arc

Areas match because speed changes.

Credit: cococubed.com

Visually reinforce equal areas before the angular momentum link.

The Ice Skater (Again)

Arms in: spins faster

(smaller \(r\))

Arms out: spins slower

(larger \(r\))

Conservation of angular momentum.

Same physics for planets: closer is faster, farther is slower.

26–27 min. “Kepler described the pattern. Newton explained the mechanism.”

Quick Check: Second Law Explained

Newton explained Kepler’s Second Law (equal areas) as a consequence of:

• The inverse-square nature of gravity
• Conservation of angular momentum because gravity is a central force
• The fact that planets have elliptical orbits
• Energy conservation

1 min. iClicker poll.

Explaining Kepler III: \(P^2 \propto a^3\)

Question: Why this exact relationship?

Newton’s Answer:

For a force \(F \propto 1/r^n\), the period-distance relationship is:

\[P^2 \propto a^{n+1}\]

• If \(n = 2\) (inverse-square): \(P^2 \propto a^3\) (matches Kepler)
• If \(n = 1\): \(P^2 \propto a^2\) (does not match)
• If \(n = 3\): \(P^2 \propto a^4\) (does not match)

27–28 min. “Only inverse-square matches Kepler’s observations!”

Newton’s Improved Version

\[P^2 = \frac{4\pi^2}{G(M + m)} a^3\]

What’s new compared to Kepler:

• The constant includes \(G\) and the total mass
• Works for any gravitational system, not just our Solar System
• We can solve for mass!

28–29 min. “This is the mass-measuring power tool.”

Motion Reveals Mass

Rearranging Newton’s version:

\[M \approx \frac{4\pi^2 a^3}{G P^2}\]

Measure orbital period and distance. Calculate central mass.

This is how we “weigh” the Sun, black holes, and galaxies.

We never touch them — we just watch how things move around them.

29–31 min. “This technique will come up again and again.”

Quick Check: Weighing the Sun

How do astronomers determine the mass of the Sun?

• By measuring how bright it is
• By measuring Earth’s orbital period and distance
• By sending a spacecraft to collect a sample
• By measuring the Sun’s size

1 min. iClicker poll.

Quick Check: Different Stars

Two planets orbit at the same distance. Star A: \(2M_\odot\). Star B: \(0.5M_\odot\). Which planet has the shorter orbital period?

• Planet A
• Planet B
• Both have the same period
• Cannot tell without planet masses

1–2 min. iClicker poll, then explain. More massive star: stronger gravity, faster orbit, shorter period.

Part 4: Synthesis

From empirical to physical

From Kepler to Newton

Kepler (Empirical)	Newton (Physical)
Describes what planets do	Explains why they do it
Patterns from data	Mechanisms from principles
Limited to observed systems	Universal — applies everywhere
Can predict within Solar System	Can predict anywhere gravity acts

31–33 min. “This is the transition from pattern to physics.”

One Equation Rules All

\[F = \frac{Gm_1m_2}{r^2}\]

With this single equation, Newton explained:

• Why apples fall
• Why the Moon orbits Earth
• Why planets orbit the Sun
• Why all three of Kepler’s laws hold
• How to weigh objects we’ll never touch

33–34 min. “One of the greatest unifications in the history of science.”

Key Takeaways (1/2)

1. Newton’s Three Laws: (1) Inertia, (2) \(F = ma\), (3) Action-Reaction

2. Circular motion = acceleration (direction changes), requiring centripetal force

3. Universal Gravitation: \(F = GMm/r^2\) — inverse-square law

4. Mass vs. Weight: Mass is intrinsic; weight depends on local gravity

34–36 min. Rapid recap.

Key Takeaways (2/2)

5. Astronauts float because of free fall, not because gravity is absent.

6. Newton explains Kepler:

   • Ellipses: inverse-square force implies conic sections
   • Equal areas: central force implies angular momentum conservation
   • \(P^2 \propto a^3\): inverse-square gives the exponent

7. Motion reveals mass: \(M \approx 4\pi^2 a^3/(G P^2)\)

The Course Connection

Motion reveals mass — the foundation for nearly everything else:

• Stars: Binary star orbits, stellar masses
• Galaxies: Rotation curves, dark matter
• Black holes: Orbiting stars, invisible mass
• Exoplanets: Stellar wobbles, planet detection

36–38 min. “We’ll use Newton’s insight over and over.”

Coming Up: Light as Information

Next lecture: Light is our only messenger from the cosmos.

We’ll learn to decode what light tells us:

• How fast is it moving? (wavelength)
• How hot is it? (spectrum)
• What is it made of? (spectral lines)

Next Steps

Reading: Lecture 7 Reading Companion

Demo: Binary Orbits (motion reveals mass)

../../../demos/binary-orbits/

38–40 min. Preview the light physics sequence (L7-L10).

Questions?

Kepler gave us the patterns.

Newton gave us the physics.

Next: Light gives us the information.

Leave time for questions before optional appendices.

Appendix: Optional Numericals

Use as backup or post-lecture reference.

Numerical Verification

Newton’s claim: the Moon is falling

Let’s verify Newton’s revolutionary claim with actual numbers.

The Moon IS Falling: Calculation

Newton’s claim: The same gravity that pulls apples pulls the Moon.

Test: Calculate the Moon’s centripetal acceleration and compare to surface gravity.

Given data:

• Earth-Moon distance: \(r = 384,400\) km \(= 3.844 \times 10^8\) m
• Moon’s orbital period: \(P = 27.3\) days
• Earth’s radius: \(R_E = 6,371\) km
• Surface gravity: \(g = 9.81\ \mathrm{m/s^2}\)

“Let’s verify Newton’s claim quantitatively. This is the calculation that unified heaven and Earth.”

Step 1: Find the Moon’s Orbital Speed

The Moon travels a circular path of circumference \(2\pi r\) in time \(P\).

Convert period to seconds:

\[P = 27.3 \text{ days} \times \frac{24 \text{ hr}}{1 \text{ day}} \times \frac{3600 \text{ s}}{1 \text{ hr}} = 2.36 \times 10^6 \text{ s}\]

Calculate orbital speed:

\[v = \frac{2\pi r}{P} = \frac{2\pi \times 3.844 \times 10^8 \text{ m}}{2.36 \times 10^6 \text{ s}}\]

\[v = \frac{2.415 \times 10^9 \text{ m}}{2.36 \times 10^6 \text{ s}} = 1,023 \text{ m/s} \approx 1.0 \text{ km/s}\]

“The Moon moves at about 1 km/s. Not that fast by space standards, but remember it’s very far away.”

Step 2: Calculate Centripetal Acceleration

\[a_c = \frac{v^2}{r} = \frac{(1,023 \text{ m/s})^2}{3.844 \times 10^8 \text{ m}}\]

\[a_c = \frac{1.047 \times 10^6 \text{ m}^2/\text{s}^2}{3.844 \times 10^8 \text{ m}} = 2.72 \times 10^{-3} \text{ m/s}^2\]

Unit check: \(\frac{\text{m}^2/\text{s}^2}{\text{m}} = \frac{\text{m}}{\text{s}^2}\) (acceleration)

The Moon’s acceleration toward Earth is \(a_c = 0.00272\ \mathrm{m/s^2}\).

Compare to surface gravity: \(g = 9.81\ \mathrm{m/s^2}\).

“Always check your units. If they don’t work out to m/s^2 for acceleration, you made a mistake.”

Step 3: Test the Inverse-Square Law

If gravity follows \(1/r^2\), then at the Moon’s distance:

\[\frac{a_{\text{Moon}}}{g_{\text{surface}}} = \frac{1}{(\text{distance ratio})^2}\]

Calculate distance ratio:

\[\frac{r_{\text{Moon}}}{R_E} = \frac{3.844 \times 10^8 \text{ m}}{6.371 \times 10^6 \text{ m}} = 60.3\]

The Moon is 60.3 Earth radii away.

Predicted acceleration ratio:

\[\frac{a_{\text{Moon}}}{g} = \frac{1}{(60.3)^2} = \frac{1}{3636} = 2.75 \times 10^{-4}\]

“Now we compare prediction to measurement.”

Step 4: Compare Prediction to Measurement

Predicted (from inverse-square):

\[a_{\text{predicted}} = \frac{g}{3636} = \frac{9.81}{3636} = 2.70 \times 10^{-3} \text{ m/s}^2\]

Measured (from orbital motion):

\[a_{\text{measured}} = 2.72 \times 10^{-3} \text{ m/s}^2\]

Match! Difference < 1%

The same gravity that pulls apples pulls the Moon.

This was Newton’s triumph: unifying terrestrial and celestial physics.

“Newton showed that there’s no separate ‘celestial physics.’ The same laws work everywhere.”

Verifying ISS “Weightlessness”

Claim: At ISS altitude (400 km), gravity is still ~90% of surface value.

Calculate:

\[r_{\text{ISS}} = R_E + h = 6,371 + 400 = 6,771 \text{ km}\]

\[\frac{g_{\text{ISS}}}{g_{\text{surface}}} = \left(\frac{R_E}{r_{\text{ISS}}}\right)^2 = \left(\frac{6,371}{6,771}\right)^2\]

\[= (0.941)^2 = 0.885 = 88.5\%\]

Gravity at ISS altitude is 88.5% of surface gravity!

Astronauts float because they’re falling, not because gravity is gone.

“400 km sounds high, but it’s only 6% farther from Earth’s center. Gravity barely decreases.”

Weighing the Sun: Full Calculation

Given:

• Earth’s orbital period: \(P = 365.25\) days \(= 3.156 \times 10^7\) s
• Earth’s orbital radius: \(a = 1.496 \times 10^{11}\) m (1 AU)
• \(G = 6.674 \times 10^{-11}\ \mathrm{N\,m^2/kg^2}\)

From Newton’s form of Kepler III:

\[M_\odot = \frac{4\pi^2 a^3}{G P^2}\]

\[M_\odot = \frac{4\pi^2 (1.496 \times 10^{11})^3}{(6.674 \times 10^{-11})(3.156 \times 10^7)^2}\]

“Let’s compute this step by step with careful unit tracking.”

Weighing the Sun: Unit Analysis

\[M_\odot = \frac{4\pi^2 a^3}{G P^2}\]

Numerator units:

\[[\text{m}^3] \quad \text{(since } a^3 \text{ is in meters cubed)}\]

Denominator units:

\[\left[\frac{\text{N} \cdot \text{m}^2}{\text{kg}^2}\right] \times [\text{s}^2] = \left[\frac{\text{kg} \cdot \text{m/s}^2 \cdot \text{m}^2}{\text{kg}^2}\right] \times [\text{s}^2] = \frac{\text{m}^3}{\text{kg}}\]

Result units:

\[\frac{\text{m}^3}{\text{m}^3/\text{kg}} = \text{kg}\]

“The units work out to kg, which is what we want for mass. Always check!”

Weighing the Sun: Result

Calculating numerator:

\[4\pi^2 (1.496 \times 10^{11})^3 = 4\pi^2 \times 3.348 \times 10^{33} = 1.322 \times 10^{35} \text{ m}^3\]

Calculating denominator:

\[(6.674 \times 10^{-11})(3.156 \times 10^7)^2 = 6.674 \times 10^{-11} \times 9.96 \times 10^{14}\] \[= 6.65 \times 10^{4} \text{ m}^3/\text{kg}\]

Final result:

\[M_\odot = \frac{1.322 \times 10^{35}}{6.65 \times 10^{4}} = 1.99 \times 10^{30} \text{ kg}\]

The Sun’s mass is \(2 \times 10^{30}\) kg — determined just by watching Earth orbit!

“We weighed an object 150 million km away, using only geometry and timing.”

Deeper Dive

Escape velocity and orbital energy

Optional advanced content.

Escape Velocity

Question: How fast must you launch an object so it never falls back?

Energy conservation:

\[\frac{1}{2}mv^2 - \frac{GMm}{r} = 0\]

(Kinetic energy at launch = gravitational potential energy to escape)

Solving for escape velocity:

\[v_{\text{escape}} = \sqrt{\frac{2GM}{r}}\]

“This is the minimum speed to escape a gravitational well.”

Escape Velocity in One Picture

Add energy and the orbit stretches.

Add enough and the path becomes unbound.

Credit: cococubed.com

Shows orbit raising and escape trajectory visually.

Earth’s Escape Velocity: Calculation

\[v_{\text{escape}} = \sqrt{\frac{2GM_E}{R_E}}\]

Given:

• \(G = 6.674 \times 10^{-11}\ \mathrm{N\,m^2/kg^2}\)
• \(M_E = 5.97 \times 10^{24}\) kg
• \(R_E = 6.371 \times 10^6\) m

\[v_{\text{escape}} = \sqrt{\frac{2 \times 6.674 \times 10^{-11} \times 5.97 \times 10^{24}}{6.371 \times 10^6}}\]

\[= \sqrt{\frac{7.97 \times 10^{14}}{6.371 \times 10^6}} = \sqrt{1.25 \times 10^8} \text{ m}^2/\text{s}^2\]

\[v_{\text{escape}} = 11,200 \text{ m/s} = \boxed{11.2 \text{ km/s}}\]

“To escape Earth, you need 11.2 km/s — about 40,000 km/hr or Mach 33.”

Escape Velocities Across the Solar System

Object	Mass (Earth = 1)	Radius (Earth = 1)	\(v_{\text{escape}}\)
Moon	0.012	0.27	2.4 km/s
Earth	1	1	11.2 km/s
Jupiter	318	11.2	59.5 km/s
Sun	333,000	109	618 km/s

“Escape velocity depends on the ratio M/R, not mass alone.” Notice: \(v_{\text{escape}} \propto \sqrt{M/R}\). Jupiter is massive but also large, so the increase is moderate.

Circular Orbital Velocity

For a circular orbit, set gravitational force = centripetal force:

\[\frac{GMm}{r^2} = \frac{mv^2}{r}\]

Solving for orbital velocity:

\[v_{\text{circular}} = \sqrt{\frac{GM}{r}}\]

Key relationship:

\[v_{\text{escape}} = \sqrt{2} \times v_{\text{circular}}\]

To escape, you need only 41% more speed than circular orbit speed!

“This is why it’s relatively easy to send probes to other planets once you’re in orbit.”