We know that gravity is all around us. Anything we throw up comes down eventually due to the gravitational pull of the Earth. But, What is escape velocity? force? Like for example, the rockets that fly into outer space defy this gravitational force. How is this possible? This is made possible due to the speed of the rocket, which exceeds a value known as escape velocity. In this article, we discuss the definition, formula, and factors affecting escape velocity.
What is escape velocity?
Escape velocity is the speed required to escape the gravitational influence of a large object. This speed enables us to generate a force that will overcome the gravitational pull of the object to a point of no return. In reality, according to the formula of gravity, where the gravitational force is inversely proportional to the square of the distance, there will be some gravitational force always, but it will not be able to pull the object back when it exceeds the escape velocity.
Escape Velocity of Earth
Image by Free-Photos from Pixabay
The escape velocity of the earth for an object is the minimum speed required to escape the gravitational pull of the planet Earth. The escape velocity of Earth is around 11.2 km/s. This is a humongous value. That is why when you hurl a ball or something into the air, it doesn’t fly into outer space. The gravity will eventually pull it down. But, the rockets fly at an extremely high speed to escape the gravitational pull of the earth. The speed is calculated so as to exceed the value of escape velocity.
At this escape velocity, the gravitational potential energy will be equal to the kinetic energy of the object. Beyond this velocity, the kinetic energy will exceed the gravitational potential energy.
Escape velocity depends on the mass of the object exerting gravitational force. So the escape velocity of the Earth will be much larger than the moon. Now, when a spacecraft returns from the moon to Earth, it can travel at a slower speed compared to its initial speed while leaving Earth. On the other hand, bigger planets will require higher gravitational force.
Earth’s atmosphere consists of many gases. It is due to the escape velocity that the molecules of a gas are held in the atmosphere. Their average velocity is less than the escape velocity of Earth.
The escape velocity of Earth, when compared to the speed of sound, is 33 times. But this value is the speed at the surface of the Ear, as we move away, this value decreases continuously.
Since the Earth is an ellipsoid, we can assume that the escape velocity at the equator might be slightly less as compared to the escape velocity at the poles.
The escape velocity of the Sun
The escape velocity of the sun is the velocity required to overcome the gravitational pull of the sun. The escape velocity of the Sun is around 618 Km/s. It is due to this high escape velocity that the planets do not drift off their orbits. For example, the velocity of the Earth around the Sun is around 30 Km/s which is way less compared to the escape velocity of the Sun.
The escape velocity of the moon
Image by WikiImages from Pixabay
The escape velocity of the moon is the velocity required to overcome the gravitational pull of the moon. The escape velocity of the moon is around 2.38 Km/s. The famous Apollo 11 rocket which reached the moon in 1969 had a speed of around 11 Km/s. This is higher than the escape velocity of the moon by a good margin to ensure the return of the spacecraft.
Escape Velocity Formula
Escape velocity is achieved when gravitational potential energy is equal to kinetic energy.
The formula for kinetic energy = 1/2(mv2)
The formula for gravitational potential energy = (GMm)/r
Where v is the escape velocity, G is the universal gravitational constant (G = 6.673 × 10^-11 Nm2 / kg2), M is the mass of the larger object, and m is the mass of the escaping smaller object.
Equating both equations we get the formula for escape velocity is:
Vc = sqrt(2GM/r)
You can use this calculator to calculate the escape velocity of any planet.
What are the factors on which the escape velocity depends?
The formula for escape velocity is Vc = sqrt(2GM/r). So, the escape velocity depends on the mass of the object exerting gravitational force and on the distance between the objects. The escape velocity is independent of the mass of the escaping object.
We can also see that the larger the mass of the object exerting gravitational force, the larger will the escape velocity. That is why the escape velocity of Sun>Earth>Moon.
The escape velocity depends on the altitude. So, the farther the object lesser will the velocity required to escape.
Calculation of the escape velocity of Earth
Formula for escape velocity = sqrt(2GM/r)
The value of G = 6.673 × 10^-11 Nm2 / kg2
R = radius of the Earth = 6.4 × 10^6 m
M = mass of the Earth = 5.972 × 10^24 kg
So escape velocity of earth = sqrt(2*6.673 × 10^-11*5.972 × 10^24/6.4 × 10^6)
Escape velocity of Earth = 11,186 m/s = ~11.2 Km/s
Difference between orbital velocity and escape velocity
| Orbital velocity | Escape velocity |
| Orbital velocity is the velocity at which an object revolves around a larger object exerting gravitational force. | Escape velocity is the velocity at which an object is able to escape the gravitational influence of a larger object. |
| It occurs when the gravitational force is equal to the centripetal force. | It occurs when the kinetic energy is equal to the gravitational potential energy. |
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Relationship between orbital velocity and escape velocity
It is clear from the above equation that the escape velocity is always greater than the value of orbital velocity. You can read this article to learn more about this.
How rockets escape the gravitational force of Earth?
The speed of the rockets is generally calculated to exceed the escape velocity of the Earth. The weight of the rockets is generally manipulated so as to adjust the speed at the same time not compromising on fuel. In a manned mission, like Apollo 11, the spacecraft has to take off from the target planet also. So, a large amount of fuel has to be carried. This makes the rocket heavy while taking off from the Earth.
What is the escape velocity of a black hole?
The name black hole is self-explanatory in that it is black and has no light. Light cannot escape out from its’s surface The escape velocity of a black hole in some points exceed the speed of light. You can read this article for more information.
Comparision of various Escape Velocities
We know that the escape velocity is dependent on the mass of the object. The higher the mass higher will be the escape velocity.
Following is the table with an escape velocity of the Sun, moon, and the Earth.
| Body | Mass (Kg) | Escape Velocity (km/s) | Comparison to speed of sound |
| Sun | 1.99 * 10^27 kg | 618 km/s | ~1802 |
| Earth | 5.972 × 10^24 kg | 11.2 km/s | ~33 |
| Moon | 7.35 * 10^22 kg | 2.38 km/s | ~7 |
Note that the escape velocity is calculated without any consideration of air resistance or any interference from foreign objects. Actual escape velocity may differ slightly based on local conditions. All calculations are made under the assumption of a spherically symmetric object, so the values may vary also based on the location of the body.
Can a bullet fire escape the gravitational pull of the earth?
The speed of a bullet is generally 0.8 m/s. At the same time, the escape velocity of the Earth is almost 11 times greater at around 11.2 Km/s. So a bullet can never escape the gravitational pull of the Earth.
Numerical question on escape velocity
Question– Suppose an asteroid is approaching the Earth(Mass = 5.972 × 10^24 kg). The center-to-center distance between the asteroid and the Earth is 24,000 Km. Find the escape velocity of the planet to escape the gravitational field of Earth so as to avoid a collision.
The formula for escape velocity (Vc) = sqrt(2GM/r)
Vc = sqrt(2*6.673 × 10^-11* 5.972 × 10^24/24,000)
Vc = 1.816 *10^5 m/s
If the asteroid travels with a velocity greater than 1.816 *10^5 m/s, then it can avoid the gravitational field of the Earth.
Question: Calculate the approximate velocity for the moon required to escape the earth’s gravitational pull
Mass of Earth = 5.972 × 10^24 kg
Distance between Earth and moon = ~384,400 km
The formula for escape velocity (Vc) = sqrt(2GM/r)
Vc = sqrt(2*6.673 × 10^-11* 5.972 × 10^24/384,400)
Vc = ~4.5*10^4 m/s
So the moon has to travel at a speed of ~4.5*10^4 m/s, to escape the gravitational pull of the Earth. The current speed of the moon is 1023 m/s. So, the moon has to increase its speed by 44 times (roughly) to escape the gravitational pull of the Earth.
Question: Calculate the approximate escape velocity of Earth to escape the sun’s gravitational pull
Mass of Sun = 1.989 × 10^30 kg
Distance between Sun and Earth = ~147220000 km
The formula for escape velocity (Vc) = sqrt(2GM/r)
Vc = sqrt(2*6.673 × 10^-11* 1.989 × 10^30/147220000)
Vc = ~13.42 *10^5 m/s
So the Earth has to travel at a speed of ~13.42 *10^5 m/s to escape the gravitational pull of the Sun. The current speed of the Earth is 460 m/s. So, the Earth has to move at 3000 times (roughly) the current speed to get off the sun’s pull. (Which looks impossible!!)
