Orbital Speed

An object in orbit is constantly being pulled toward the body it orbits. Rather than escaping gravity, an orbiting object is actually in continuous free fall.

For a circular orbit, gravity supplies exactly the inward acceleration needed to keep the object moving along its curved path.

The circular orbital speed is:

v = √(GM/r)

where:

• v = orbital speed in m/s
• G = gravitational constant = 6.67 × 10⁻¹¹ N·m²/kg²
• M = mass of the central body in kg
• r = orbital radius measured from the centre of the central body in m

The equation shows that orbital speed depends on the mass of the central body and the orbital radius, but not on the mass of the satellite.

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Orbiting Objects Are Falling

Imagine throwing a ball horizontally from a high mountain.

Gravity pulls the ball downward, so it follows a curved path and eventually strikes Earth.

If the ball is thrown faster:

• it travels farther horizontally
• Earth curves away beneath it
• it takes longer to reach the surface

If it could be launched sufficiently fast and atmospheric resistance were ignored, the curvature of its falling path could match the curvature of Earth.

The ball would continue falling without reaching the surface.

It would be in orbit.

This idea is often called Newton's cannonball thought experiment.

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Continuous Free Fall

A satellite in orbit experiences gravitational acceleration.

Therefore, it is continuously falling toward Earth.

However, it also has a large sideways velocity.

During the time the satellite falls toward Earth, the curved surface of Earth falls away beneath it.

The satellite therefore continually misses Earth.

This produces an orbit.

The important idea is:

An orbit is not the absence of falling. An orbit is a particular kind of continuous falling.

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Gravity Does Not Disappear in Orbit

A common misconception is that satellites orbit because they are beyond Earth's gravity.

They are not.

Earth's gravity extends indefinitely, becoming weaker with increasing distance.

At the altitude of many artificial satellites, gravity is still quite strong.

Without Earth's gravity, a satellite would not remain in orbit.

It would move in approximately a straight line according to Newton's First Law.

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Gravity Bends the Path

At any instant, a satellite's velocity is approximately tangent to its circular orbit.

Gravity acts toward Earth's centre.

Therefore:

• velocity points sideways
• gravitational force points inward
• gravitational acceleration points inward

Gravity continuously changes the direction of the velocity.

This bends the satellite's path into a circle.

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Constant Speed but Changing Velocity

For an ideal circular orbit:

speed is constant

but:

velocity is continuously changing

This happens because velocity includes both:

• magnitude
• direction

Even if the magnitude remains constant, the direction changes at every point around the orbit.

Therefore, the satellite is accelerating.

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Centripetal Acceleration

Any object moving in a circle requires an inward acceleration called centripetal acceleration.

Its magnitude is:

a_c = v²/r

For a satellite, gravity supplies this acceleration.

Gravitational field strength at distance r is:

g = GM/r²

Therefore, for a circular orbit:

a_c = g

or:

v²/r = GM/r²

This relationship allows orbital speed to be calculated.

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Deriving Circular Orbital Speed

The gravitational force on a satellite of mass m is:

F_gravity = GMm/r²

The required centripetal force is:

F_c = mv²/r

For a circular orbit, gravity provides the centripetal force:

GMm/r² = mv²/r

Cancel the satellite mass m:

GM/r² = v²/r

Multiply by r:

GM/r = v²

Take the square root:

v = √(GM/r)

This is the equation for circular orbital speed.

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Why Satellite Mass Does Not Matter

Notice that the satellite mass m cancels during the derivation.

A larger satellite experiences a larger gravitational force:

F = GMm/r²

but it also requires proportionally more force to produce the same acceleration:

F = ma

Therefore, satellites of different masses can orbit at the same speed if they are at the same orbital radius around the same central body.

A 10 kg satellite and a 10,000 kg satellite would require the same ideal circular orbital speed at the same location.

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What Determines Orbital Speed?

From:

v = √(GM/r)

orbital speed depends on two quantities:

• mass of the central body, M
• orbital radius, r

For satellites around the same planet, M remains constant.

Therefore:

v ∝ 1/√r

This means orbital speed decreases as orbital radius increases.

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Orbital Radius

Orbital radius is measured from the centre of the planet, not from its surface.

Therefore:

r = R + h

where:

• R = radius of the planet
• h = altitude above the surface

This distinction is essential in orbital calculations.

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Worked Example: Finding Orbital Radius

A satellite is:

600 km above Earth's surface

Earth's radius is approximately:

6370 km

Therefore:

r = 6370 + 600

= 6970 km

Convert to metres:

r = 6.97 × 10⁶ m

This is the value that must be used in the orbital-speed equation.

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Calculating Orbital Speed

Suppose a satellite orbits Earth at:

r = 7.00 × 10⁶ m

Use:

G = 6.67 × 10⁻¹¹ N·m²/kg²

M_Earth = 5.97 × 10²⁴ kg

Then:

v = √(GM/r)

v = √[(6.67 × 10⁻¹¹)(5.97 × 10²⁴)/(7.00 × 10⁶)]

v ≈ √(5.69 × 10⁷)

v ≈ 7.54 × 10³ m/s

Therefore:

v ≈ 7.5 km/s

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Low Earth Orbit

Many satellites orbit relatively close to Earth in low Earth orbit (LEO).

Typical orbital speeds are around:

7–8 km/s

That means a satellite can travel thousands of kilometres in only a few minutes.

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Example: Satellite 400 km Above Earth

Suppose:

h = 400 km

Earth radius:

R = 6370 km

Therefore:

r = 6370 + 400

= 6770 km

= 6.77 × 10⁶ m

Now:

v = √[(6.67 × 10⁻¹¹)(5.97 × 10²⁴)/(6.77 × 10⁶)]

v ≈ 7.67 × 10³ m/s

Therefore:

v ≈ 7.7 km/s

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Orbital Speed and Altitude

For satellites orbiting the same planet:

v ∝ 1/√r

Therefore:

higher orbit → lower orbital speed

and:

lower orbit → higher orbital speed

This may seem surprising because higher satellites have farther to travel around their larger orbits.

However, gravity is weaker at greater distances.

Less centripetal acceleration is required, so the satellite travels more slowly.

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Comparing Two Orbital Radii

Suppose Satellite A orbits at radius:

r

Satellite B orbits at:

4r

Since:

v ∝ 1/√r

then:

vB/vA = √(r/4r)

= 1/2

Therefore:

vB = 0.5vA

A satellite orbiting four times farther from the centre moves at half the circular orbital speed.

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Another Comparison

Suppose Satellite A orbits at:

r

Satellite B orbits at:

9r

Then:

vB/vA = √(1/9)

= 1/3

Satellite B moves at one-third the orbital speed of Satellite A.

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Orbital-Speed Pattern

For satellites around the same central body:

Orbital speed follows an inverse square-root relationship with radius.

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Comparing Satellites at Different Altitudes

Consider two satellites around Earth.

Satellite A is in a low orbit.

Satellite B is much farther from Earth.

Satellite A experiences:

• stronger gravity
• greater centripetal acceleration
• greater orbital speed

Satellite B experiences:

• weaker gravity
• smaller centripetal acceleration
• lower orbital speed

However, Satellite B travels around a much larger orbital path.

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Higher Orbits Also Have Longer Periods

Orbital period T is the time required to complete one orbit.

For a circular orbit:

T = 2πr/v

A higher satellite has:

• larger r
• smaller v

Both effects increase the orbital period.

Therefore:

higher orbit → longer orbital period

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Worked Example: Orbital Period

Suppose:

r = 7.0 × 10⁶ m

v = 7.5 × 10³ m/s

Then:

T = 2πr/v

= 2π(7.0 × 10⁶)/(7.5 × 10³)

≈ 5860 s

Convert to minutes:

5860/60 ≈ 97.7 min

So the satellite completes approximately one orbit every:

98 minutes

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Kepler's Third Law

Combining:

v = √(GM/r)

with:

T = 2πr/v

gives:

T = 2π√(r³/GM)

Therefore:

T² ∝ r³

for objects orbiting the same central body.

This relationship is known as Kepler's Third Law.

It shows that orbital period increases rapidly as orbital radius increases.

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Low Orbit Versus Geostationary Orbit

A low-Earth-orbit satellite may:

• travel at roughly 7–8 km/s
• complete an orbit in roughly 90–100 minutes

A geostationary satellite travels much farther from Earth.

Its altitude is approximately:

35,786 km

Its orbital speed is only about:

3.1 km/s

and its orbital period matches Earth's rotation.

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Geostationary Satellites

A geostationary satellite appears to remain above approximately the same point on Earth's equator.

To achieve this, it must:

• orbit above the equator
• move in the same direction as Earth's rotation
• have an approximately circular orbit
• have an orbital period equal to Earth's rotational period

This requires a specific orbital radius.

If the satellite were significantly closer to Earth, it would orbit too quickly.

If it were significantly farther away, it would orbit too slowly.

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Why Higher Orbits Are Slower

We can understand this from the force equation.

Gravitational force decreases as:

1/r²

At larger r, gravity provides less inward acceleration.

Therefore, the satellite does not need to move as quickly for gravity to curve its path into the required circle.

Mathematically:

v²/r = GM/r²

so:

v² = GM/r

and:

v = √(GM/r)

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Orbital Speed Around Other Bodies

The same equation works for any approximately spherical gravitational source:

v = √(GM/r)

This includes:

• Earth
• Moon
• Mars
• Jupiter
• Sun
• other stars

Only M and r change.

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Orbiting the Moon

Suppose a spacecraft orbits the Moon at:

r = 2.00 × 10⁶ m

Moon mass:

M = 7.35 × 10²² kg

Then:

v = √[(6.67 × 10⁻¹¹)(7.35 × 10²²)/(2.00 × 10⁶)]

v ≈ 1.57 × 10³ m/s

Therefore:

v ≈ 1.6 km/s

This is much lower than a typical low-Earth orbital speed.

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Why Lunar Orbital Speeds Are Lower

The Moon has much less mass than Earth.

Therefore, its gravitational field is weaker at comparable relative distances.

Less centripetal acceleration is required for a stable circular orbit.

This results in lower orbital speeds.

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Earth Orbiting the Sun

Earth itself is an orbiting object.

For a simplified circular model:

v = √(GM_Sun/r)

Earth travels around the Sun at approximately:

30 km/s

This means Earth moves almost 30 kilometres through its orbit every second.

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The Same Physics Applies

The same basic relationship explains:

• artificial satellites around Earth
• the Moon around Earth
• planets around the Sun
• moons around planets
• stars orbiting within binary systems

Gravity provides the inward acceleration that continually changes the direction of motion.

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Analyzing Satellite Motion

Consider a satellite in a circular orbit.

At any instant:

velocity points tangent to the orbit

while:

gravity points toward the centre

The satellite would naturally move in a straight line because of inertia.

Gravity continually deflects it from that straight path.

The combination creates orbital motion.

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What If Gravity Suddenly Disappeared?

Imagine Earth's gravity suddenly vanished.

The satellite would no longer curve around Earth.

According to Newton's First Law, it would continue with its instantaneous velocity.

Therefore, it would move along a line tangent to its former orbit.

This demonstrates that gravity is responsible for continually bending the path.

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What If the Satellite Slows Down?

Suppose a satellite is initially in a circular orbit and suddenly loses some forward speed.

It no longer has the correct speed for that circular orbit.

Gravity bends its trajectory more strongly than required to maintain the original circle.

The satellite enters a new orbit with a lower point.

If enough speed is lost, the new trajectory may enter the atmosphere or intersect the planet.

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What If the Satellite Speeds Up?

Suppose a satellite receives a forward engine burn.

Its kinetic energy increases.

It no longer follows its original circular orbit.

Instead, it generally enters an elliptical orbit with a higher point on the opposite side.

If sufficient energy is added, it may eventually reach escape conditions.

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Circular Orbit Is a Specific Condition

At a particular radius, there is a particular speed required for an ideal circular orbit:

v = √(GM/r)

Too little speed:

the object falls into a lower trajectory.

Correct speed:

circular orbit.

Greater speed but below escape conditions:

typically an elliptical bound orbit.

At sufficient speed:

the object can escape.

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Circular and Elliptical Orbits

Not all satellite orbits are circular.

Many are elliptical.

In an elliptical orbit:

• orbital radius changes
• gravitational potential energy changes
• kinetic energy changes
• orbital speed changes

The circular orbital-speed equation cannot simply be used at every point of an elliptical orbit.

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Speed in an Elliptical Orbit

An object in an elliptical orbit moves:

fastest when closest to the central body

and:

slowest when farthest away

The closest point is called periapsis.

The farthest point is called apoapsis.

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Energy Changes in an Elliptical Orbit

Near periapsis:

• gravitational potential energy is lower
• kinetic energy is greater
• speed is greater

Near apoapsis:

• gravitational potential energy is higher
• kinetic energy is smaller
• speed is lower

Mechanical energy remains constant if no non-conservative forces do work.

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Energy in a Circular Orbit

For a circular orbit:

Eₖ = 1/2 mv²

Since:

v² = GM/r

then:

Eₖ = GMm/(2r)

Gravitational potential energy is:

Eₚ = −GMm/r

Therefore:

Eₚ = −2Eₖ

Total energy is:

E_total = Eₖ + Eₚ

so:

E_total = −GMm/(2r)

The negative total energy shows that the satellite is gravitationally bound.

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Higher Orbit and Energy

A satellite in a higher circular orbit moves more slowly.

However, its total mechanical energy is actually greater because it is less negative.

For a circular orbit:

E_total = −GMm/(2r)

As r increases:

E_total approaches zero.

Therefore, moving a satellite into a higher circular orbit requires energy even though its final orbital speed is lower.

This is one of the counterintuitive features of orbital mechanics.

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How Can Adding Energy Make the Final Speed Lower?

Suppose a spacecraft fires its engine forward.

Initially:

• speed increases
• kinetic energy increases

This places the spacecraft into an elliptical transfer orbit.

As it climbs:

• gravitational potential energy increases
• kinetic energy decreases
• speed decreases

After another manoeuvre establishes a higher circular orbit, the final orbital speed can be lower than the original low-orbit speed.

Yet the spacecraft has more total mechanical energy.

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Orbital Speed and Escape Speed

At the same radius:

v_orbit = √(GM/r)

Escape speed is:

v_escape = √(2GM/r)

Therefore:

v_escape = √2 v_orbit

Since:

√2 ≈ 1.414

escape speed is approximately:

1.414 times circular orbital speed

at the same location.

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Worked Example

A satellite has circular orbital speed:

7.2 km/s

The escape speed at that radius is:

v_escape = √2(7.2)

≈ 10.2 km/s

Therefore, the satellite is moving below the local escape speed and remains gravitationally bound.

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Astronauts and Apparent Weightlessness

Astronauts aboard an orbiting spacecraft experience apparent weightlessness.

This is not because gravity has disappeared.

The spacecraft, astronauts, and objects inside are all falling toward Earth together.

Because the floor does not provide the normal supporting force experienced when standing on Earth, astronauts appear weightless.

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Microgravity

The term microgravity is often used to describe the environment inside an orbiting spacecraft.

Gravity itself may still be substantial.

The apparent weightlessness results mainly from continuous free fall.

Small residual accelerations can arise from:

• atmospheric drag
• spacecraft rotation
• vibrations
• tidal effects
• onboard equipment

Therefore, the environment is called microgravity rather than perfectly zero gravity.

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Atmospheric Drag and Orbital Decay

Low-orbiting satellites may encounter traces of Earth's atmosphere.

Even though the atmosphere is extremely thin at these altitudes, drag can gradually remove orbital energy.

This causes orbital decay.

As the orbit becomes lower:

• atmospheric density increases
• drag increases
• energy loss becomes faster

Eventually, the satellite may re-enter the atmosphere.

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The Orbital-Decay Paradox

Drag initially slows a satellite.

Yet a satellite in a lower circular orbit has a higher orbital speed.

This seems contradictory.

The explanation involves energy.

Drag removes total mechanical energy.

The satellite moves into a lower orbit where gravitational potential energy is more negative.

Part of that decrease in potential energy appears as kinetic energy.

Therefore, the final lower circular orbit can have a higher speed even though drag caused the orbital energy loss.

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Real Satellite Motion

Real satellites are affected by more than the simple two-body gravitational model.

Their trajectories may be influenced by:

• atmospheric drag
• Earth's non-spherical shape
• the Moon
• the Sun
• solar radiation pressure
• other planets
• engine burns

Nevertheless:

v = √(GM/r)

is an extremely useful model for approximately circular orbits.

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A Strategy for Orbital-Speed Problems

First, identify the central body's mass:

M

Then determine the orbital radius:

r = R + h

Convert all distances to metres.

Use:

v = √(GM/r)

For comparisons around the same body:

v₂/v₁ = √(r₁/r₂)

For orbital period:

T = 2πr/v

For local escape speed:

v_escape = √2 v_orbit

Finally, check whether the answer makes physical sense:

higher circular orbit should normally mean lower orbital speed.

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Common Mistakes

Thinking orbiting objects are not falling

Orbiting objects are in continuous free fall.

Thinking gravity is absent in orbit

Gravity provides the centripetal acceleration required for orbit.

Thinking constant speed means zero acceleration

The direction of velocity changes continuously, so the satellite accelerates.

Using altitude instead of orbital radius

Use:

r = R + h

Thinking higher satellites move faster

For circular orbits around the same body:

higher r → lower v

Thinking satellite mass affects orbital speed

Satellite mass cancels from the orbital-speed equation.

Treating gravity and centripetal force as two separate forces

Gravity is the force providing the required centripetal force.

Using the circular-orbit equation for an elliptical orbit

Speed changes throughout an elliptical orbit.

Thinking astronauts float because gravity is zero

They experience apparent weightlessness because they and their spacecraft are falling together.

Thinking adding energy must produce a faster final circular orbit

A higher circular orbit has greater total energy but lower orbital speed.

Confusing orbital speed with escape speed

At the same radius:

v_escape = √2 v_orbit

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Key Terms

Orbital speed — magnitude of the velocity of an object moving in orbit.

Orbital velocity — vector describing both the speed and direction of an orbiting object.

Orbit — gravitationally curved trajectory around another body.

Circular orbit — orbit with constant distance from the central body.

Free fall — motion in which gravity is the dominant force acting on an object.

Continuous free fall — description of orbital motion in which an object continually falls toward a body while moving sideways fast enough to keep missing it.

Centripetal force — net inward force required for circular motion.

Centripetal acceleration — acceleration directed toward the centre of circular motion.

Tangential velocity — velocity directed tangent to the orbital path.

Orbital radius — distance from the centre of the central body to the orbiting object.

Altitude — height above the surface of a celestial body.

Orbital period — time required to complete one orbit.

Low Earth orbit (LEO) — relatively low-altitude orbit around Earth.

Geostationary orbit — equatorial orbit with a period matching Earth's rotation so the satellite appears approximately stationary over one location.

Elliptical orbit — orbit whose distance from the central body varies.

Periapsis — point in an orbit closest to the central body.

Apoapsis — point in an orbit farthest from the central body.

Orbital decay — gradual reduction in orbital energy and altitude, often caused by drag.

Microgravity — condition of very small apparent weight experienced in an orbiting spacecraft.

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Key Takeaways

• Orbiting objects remain in continuous free fall.
• A satellite does not orbit because gravity disappears; it orbits because gravity continually bends its path.
• A satellite's velocity is tangential to its orbit while gravitational acceleration points inward.
• Gravity supplies the centripetal force required for circular motion.
• Circular orbital speed is v = √(GM/r).
• Orbital speed does not depend on satellite mass.
• Orbital radius is measured from the centre of the central body.
• For a satellite above a planet, r = R + h.
• Around the same body, orbital speed varies as 1/√r.
• Lower circular orbits have higher orbital speeds.
• Higher circular orbits have lower orbital speeds.
• Higher orbits also have longer orbital periods.
• For objects around the same central body, T² ∝ r³.
• A circular orbit has constant speed but continuously changing velocity.
• An orbiting satellite therefore has centripetal acceleration even when its speed is constant.
• Astronauts experience apparent weightlessness because they and their spacecraft are falling together.
• Elliptical orbits have changing orbital speeds.
• Objects in elliptical orbits move fastest near periapsis and slowest near apoapsis.
• A higher circular orbit has lower speed but greater total mechanical energy.
• At the same radius, v_escape = √2 v_orbit.
• Changes in satellite speed alter the size and shape of its orbit.
• Orbital speed links gravity, circular motion, energy and satellite motion into a single physical model.

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Check Your Understanding

1. Explain why a satellite in orbit is described as being in continuous free fall.
2. If gravity suddenly disappeared, what path would an orbiting satellite follow? Explain your answer.
3. Explain why a satellite can have constant speed while still accelerating.
4. Starting with GMm/r² = mv²/r, derive the circular orbital-speed equation.
5. Explain why satellite mass does not appear in v = √(GM/r).
6. A satellite orbits 500 km above Earth. Earth's radius is 6370 km. Calculate its orbital radius in metres.
7. Using M_Earth = 5.97 × 10²⁴ kg, calculate the orbital speed of the satellite in Question 6. Use G = 6.67 × 10⁻¹¹ N·m²/kg².
8. Satellite A orbits at radius r with speed v. Satellite B orbits the same planet at radius 4r. Determine Satellite B's orbital speed in terms of v.
9. Another satellite orbits at 9r. Determine its orbital speed relative to a satellite at r.
10. Explain why satellites at greater orbital radii move more slowly but take longer to complete an orbit.
11. A satellite has orbital radius 8.0 × 10⁶ m and speed 7.0 × 10³ m/s. Calculate its centripetal acceleration.
12. A satellite's circular orbital speed is 6.5 km/s. Calculate the escape speed at the same radius.
13. Explain why astronauts aboard an orbiting spacecraft appear weightless even though Earth's gravitational field is still acting strongly on them.
14. Describe what happens to the trajectory of a satellite if its forward speed is suddenly increased while remaining below escape conditions.
15. A student says, "A satellite in a higher orbit must move faster because it has farther to travel." Evaluate this statement using the relationship between orbital speed, orbital radius, gravitational field strength, and orbital period.