Orbital Velocity Formula

Orbital velocity is defined as the velocity at which a body circles around another body. The more substantial the body in the centre of attraction is, the higher the orbital velocity for a given height or distance. For an artificial or natural satellite, orbital velocity can be interpreted as the velocity necessary to maintain it in its orbit. Space organizations rely on it heavily to figure out how to launch satellites. It aids scientists in determining the speeds at which satellites must rotate around a planet or celestial body in order to avoid colliding with it.

Formula

The orbital velocity of a satellite orbiting around the Earth is determined by its height above the Earth. More is the orbital velocity, the closer satellite is to the Earth. It is equal to the square root of the product of the gravitational constant and mass of the body divided by the radius of its orbit.

where,

G is the gravitational constant,

M is the mass of object at centre,

R is the radius of the orbit.

Derivation

The formula for orbital velocity is derived through the concepts of gravitational force and centripetal force.

Suppose a satellite of mass m and radius r is orbiting circularly around planet Earth at an altitude h from Earth surface. Let us say, the mass and radius of Earth is M and R respectively. This implies that,

=> r = R + h   ……. (1)

Now, we know that to make the satellite revolve in its orbit, a centripetal force of mV²/r is required. This force is provided by the gravitational force existing between the satellite and the earth.

So, we have

=> mV²/r = GMm/r²

=> V² = GM/r

Using (1), we have

=> V² = GM/(R + h)

As (R + h) ≈ R, we get

This derives the formula for orbital velocity of an object or satellite revolving around a planet.

Sample Problems

Problem 1. Find the orbital velocity of an object revolving around the planet Earth if the radius of Earth is 6.5 × 10⁶ m, the mass of Earth is 5.9722 × 10²⁴ kg and the gravitational constant G is 6.67408 × 10^-11 m³ kg^-1 s^-2.

Solution:

We have,

G = 6.67408 × 10^-11

R = 6.5 × 10⁶

M = 5.9722 × 10²⁴

Using the formula we have,

V = √(GM/R)

= (6.67408 × 10^-11)(5.9722 × 10²⁴)/(6.5 × 10⁶)

= 29.8 km/s

Problem 2. Find the orbital velocity of an object revolving around the planet Mercury if the radius of Mercury is 2439.7 km, the mass of Mercury is 0.33 × 10²⁴ kg and the gravitational constant G is 6.67408 × 10^-11 m³ kg^-1 s^-2.

Solution:

We have,

G = 6.67408 × 10^-11

R = 2439.7

M = 0.33 × 10²⁴

Using the formula we have,

V = √(GM/R)

= (6.67408 × 10^-11)(0.33 × 10²⁴)/(2439.7)

= 47.4 km/s

Problem 3. Find the orbital velocity of an object revolving around the planet Venus if the radius of Venus is 6051.8 km, the mass of Venus is 4.87 × 10²⁴ kg and the gravitational constant G is 6.67408 × 10^-11 m³ kg^-1 s^-2.

Solution:

We have,

G = 6.67408 × 10^-11

R = 6051.8

M = 4.87 × 10²⁴

Using the formula we have,

V = √(GM/R)

= (6.67408 × 10^-11)(4.87 × 10²⁴)/(6051.8)

= 35 km/s

Problem 4. Find the orbital velocity of an object revolving around the planet Mars if the radius of Mars is 3389 km, the mass of Mars is 0.642 × 10²⁴ kg and the gravitational constant G is 6.67408 × 10^-11 m³ kg^-1 s^-2.

Solution:

We have,

G = 6.67408 × 10^-11

R = 3389

M = 0.642 × 10²⁴

Using the formula we have,

V = √(GM/R)

= (6.67408 × 10^-11)(0.642 × 10²⁴)/(3389)

= 24.1 km/s

Problem 5. Find the orbital velocity of an object revolving around the planet Jupiter if the radius of Jupiter is 69911 km, the mass of Jupiter is 1898 × 10²⁴ kg and the gravitational constant G is 6.67408 × 10-11 m3 kg-1 s-2.

Solution:

We have,

G = 6.67408 × 10^-11

R = 69911

M = 1898 × 10²⁴

Using the formula we have,

V = √(GM/R)

= (6.67408 × 10^-11)(1898 × 10²⁴)/(69911)

= 13.1 km/s

Problem 6. Find the orbital velocity of an object revolving around the planet Saturn if the radius of Saturn is 58232 km, the mass of Saturn is 568 × 10²⁴ kg and the gravitational constant G is 6.67408 × 10-11 m3 kg-1 s-2.

Solution:

We have,

G = 6.67408 × 10^-11

R = 58232

M = 568 × 10²⁴

Using the formula we have,

V = √(GM/R)

= (6.67408 × 10^-11)(568 × 10²⁴)/(58232)

= 9.7 km/s

Problem 7. Find the orbital velocity of an object revolving around the planet Uranus if the radius of Uranus is 25362 km, the mass of Uranus is 86.8 × 10²⁴ kg and the gravitational constant G is 6.67408 × 10-11 m3 kg-1 s-2.

Solution:

We have,

G = 6.67408 × 10^-11

R = 25362

M = 86.8 × 10²⁴

Using the formula we have,

V = √(GM/R)

= (6.67408 × 10^-11)(86.8 × 10²⁴)/(25362)

= 6.8 km/s