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# Orbital Velocity Formula

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Last updated date: 09th Aug 2024
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## Orbital Velocity

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The orbital velocity of a body is the velocity at which it revolves around the other body. The objects that travel around the Earth in uniform circular motion are said to be in an orbit. The velocity of this orbit depends on the distance between the object under consideration and the center of the earth. This velocity is usually associated with artificial satellites because they can revolve around any particular planet. The Orbital Velocity Formula is used to calculate the orbital velocity of an object if its mass and radius are known.

The moving body has a tendency to move in a straight path because of its inertia. The gravitational force, on the other hand, tends to drag it down. The orbital route, which is elliptical or circular in shape, is a balance of gravity and inertia. The velocity required to achieve a balance between gravity's pull on the body and the inertia of the body's motion is known as orbital velocity. The orbital velocity of a satellite orbiting around the Earth is determined by its altitude above the Earth. The faster the required orbital velocity, the closer it is to the Earth.

At lower altitudes, a satellite collides with traces of Earth's atmosphere, causing drag. This drag causes the satellite's orbit to degrade, eventually causing it to fall back into the atmosphere and burn up.

### What Is Orbital Velocity?

A natural or artificial satellite's orbital velocity is the velocity required to keep it in orbit. The moving body's inertia causes it to move in a straight line, while gravitational force pulls it down. The elliptical or circular orbital path thus shows a balance between gravity and inertia. If the muzzle velocity of a cannon shot from a mountaintop is increased, the projectile will go further. The bullet will never fall to the ground if the velocity is great enough. The Earth's surface can be imagined curving away from the projectile, or satellite, at the same rate that it falls toward it. The higher the orbital velocity for a given height or distance, the more massive the body is at the center of attraction. If air resistance could be ignored near the Earth's surface, orbital velocity would be around eight kilometers (five miles) per second. The gravitational force is weaker the farther a satellite is from the center of attraction, and the less velocity it needs to stay in orbit. See also the definition of escape velocity.

Orbital Velocity Equation

The equation of the orbital velocity is given by:

$V_{orbit} = \sqrt{\frac{GM}{R}}$

In the above equation, G stands for Gravitational Constant, M stands for the mass of the body at the center and R is the radius of the orbit.

The Orbital Velocity Equation is used to find the orbital velocity of a planet if its mass M and radius R are known.

The unit used to express Orbital Velocity is meter per second (m/s).

### Derive An Expression for Orbital Velocity

To derive the orbital velocity, we first need to know about the gravitational force and the centripetal force. It is important to know about the gravitational force because it is this force that allows orbiting to occur. This gravitational force is exerted by a central body on the orbiting body to keep it in its orbit. Centripetal force is also important because it is this force that is responsible for circular motion.

For the derivation of the formula, let us take a satellite of mass m which revolves around the Earth in a circular orbit of radius r at a height h from the surface of the Earth. Let M and R be the mass and radius of the Earth respectively, then r=R+h.

To resolve the satellite, a centripetal force $\frac{mv_{0}^{2}}{r}$ is needed which is provided by the gravitational force of the G$\tfrac{Mm}{r^{2}}$ acting between the satellite and the Earth.

Thus, equating the two equations, we got

$\frac{mv_{0}^{2}}{r}\ = G\tfrac{Mm}{r^{2}}$

$v_{0}^{2}= \frac{GM}{r}= \frac{GM}{R+h}$

Simplifying the above equation, we get

$v_{0}= \sqrt{\frac{GM}{R+h}}$................(1)

Also,GM = g$R^{2}$, where g is the acceleration due to gravity

Therefore,

$v_{0}= \sqrt{\frac{gR^{2}}{R+h}}$

Simplifying the above equation, we get

$v_{0}= R\sqrt{\frac{g}{R+h}}$

Let g’ be the acceleration due to gravity at a height h from the surface

$g^{'}= \frac{GM}{(R+h)^{2}}$

### Conclusion

Orbital speed is a very important and rewarding chapter in physics. It helps with the understanding of orbital concepts in physics. With thorough research, you can achieve very good results in this area.

## FAQs on Orbital Velocity Formula

1. What Is Orbital Velocity?

Orbital velocity refers to the velocity required by the satellites to remain in their orbits. The orbital pathway being either elliptical or circular displays a balance between the inertia of the satellite which helps to  make it move in a straight line with the gravity of the planet which in turn pulls the satellite closer and closer to the planet. This orbital velocity is also dependent on the distance between the center of the planet and the satellite revolving around it. This velocity would be higher if the center of attraction is a more massive body at a particular altitude.

2. Explain The Relationship Between Escape Velocity And Orbital Velocity In Terms of The Earth.

Rocket scientists use the principle of orbital speed to determine the course of space flight. This includes both launching rockets into the sky, establishing orbits, changing orbits, and even getting out of orbit, returning to Earth or setting a new course in space.

Where the rocket is launched depends on physics. Rockets are usually launched as close to the equator as possible  to take advantage of the Earth's rotation speed. This speed is the highest at the equator, about 1,000 mph. The higher the orbital speed that a rocket receives from the Earth, the less fuel it needs to reach that orbital speed, and the more efficient it is. Not all rockets can take advantage of the Earth's rotation. Some rockets are designed to send satellite-like payloads  into north-south orbit around the poles.

Orbital mechanics is a term used to describe the mathematics that spacecraft use to change orbit. For objects  in orbit, the closer they are to the objects in orbit, the faster they will be in orbit. This applies to all objects in different orbits, such as the Earth orbiting the Sun, the Moon orbiting the Earth, or a spacecraft orbiting a planet. In orbital mechanics, the concepts of acceleration and deceleration are complex and unintuitive. In orbit, the forward launch of the engine moves it to a higher orbit. This actually means that the slower the movement of the object in the higher orbit, the slower it will be. To be faster, you need to slow down and fall into a lower orbit.

The farther away  from the earth, the less  this effect. When  far enough  from Earth, the relative effects of orbital mechanics are so small that you can navigate the spacecraft as if you were manipulating it in  space.

3. What is Transverse Orbital Speed?

In space exploration, the idea of orbital velocity is crucial. 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 heavenly body in order to avoid colliding with it.

The two most critical calculated parameters needed to plan a space mission or a satellite mission are orbital velocity and escape velocity. In competitive exams, problems involving these two entities are frequently asked. As a result, it is critical to understand these formulas as well as the logic behind them in order to answer problems quickly.

The Formula: Let's take a closer look at the formula.

The orbital velocity of any rotating object can be calculated using the formula,

$V_{orbit} = \sqrt{\frac{GM}{R}}$

Where G is the gravitational constant, M is the mass of the body at its center, and R is the orbital radius.

M is almost always the weight of the earth.

4.How is the orbital velocity related to the rotation velocity of the earth?

The earth is always rotating around its axis. Such axes pass through the planet's North  and South Pole. The part of the earth farthest from this polar central axis is the equator. According to the principle of physics, the rotation speed of the earth is the fastest at the point farthest from the axis. For this reason, space agencies tend to launch  rockets from points  relatively close to the equator. This allows the rocket to take over the speed of rotation of the Earth in order to reach its orbital speed.

5. What Factors Affect Orbital Velocity?

The curved surface of a planet, star, or other celestial body allows for orbital velocity. An orbiting item travels in a straight line, but the body it orbits curves. As a result, assuming the orbiting item maintains the right speed, the continual curvature of the orbited body prevents the orbiting object from falling all the way to the surface.

Due to the principle of inertia, it is easier to maintain a steady speed in space than it is on Earth. An item in motion tends to continue in motion unless acted on by an outside force, according to one of Sir Isaac Newton's laws of inertia. A flying item hits many air molecules in the earth's atmosphere, which slows the object's speed as it passes through the sky. The air becomes more void as you travel beyond the earth's atmosphere into higher altitudes, with fewer molecules to oppose the forward velocity of an orbiting spacecraft.

6. Explain The Relationship Between Escape Velocity And Orbital Velocity In Terms of The Earth.

Escape velocity is the velocity required for a free, non-propelled object to break free from the gravitational impact of a huge center body whereas the orbital velocity is the velocity that is required by objects to remain in their orbits.

It takes a certain level of velocity for an object to achieve orbit around a planet such as Earth. To break free from such an orbit, an even greater velocity is required.

The escape velocity is consistently √2 times the orbital speed. At a distance from the moon to the earth, the orbital speed is 1.022 km/s. The escape velocity consequently at this separation is 1.445 km/s. However, this would need to be coordinated away from the Earth.

7. What is Transverse Orbital Speed?

The transverse orbital speed is inversely proportional to the distance to the central body by the law of conservation of angular force or Kepler’s second law. This states that as a body moves around its orbit during a fixed time, the line from the barycenter to the body sweeps a steady zone of the orbital plane, paying little heed to which part of its orbit the body traces during that time. This implies that at a smaller distance, to cover the same distance, the object needs to move faster in the arc. Hence, a body moves slower near its apoapsis than near its periapsis.

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