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When we throw an object towards the sky, it doesn’t fly through the air and escape to space. This is due to the gravitational pull. So how does a rocket escape into outer space? The space vehicle requires an immense quantity of fuel to break through the earth’s gravitational pull. This explains what is escape velocity for the earth or the escape speed. This is the minimum speed required to break free the gravitational pull. The object needs to achieve the escape velocity of the celestial bodies like natural satellites and planets. This allows for escaping the influence of the gravitational sphere of the celestial body. The sum total of kinetic energy and gravitational potential energy of the system will be zero in this given velocity.

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Escape speed is defined as the minimum speed with which a mass needs to be propelled from the earth’s surface to escape earth’s gravity. What is escape velocity for the earth is also the escape speed. This is the minimum speed needed for an object to free from the gravitation force of a massive object.

To understand what speed is escape velocity, let’s view the earth as a massive body. The escape velocity is the minimum speed or velocity that an object should gain to overcome the gravitational field of earth and travel to infinite space without falling back. It totally depends on the mass of the massive body and the distance of the object from the massive body. The more mass and closer the distance of the massive body, greater will be the escape velocity.

The derivation of escape speed is defined in terms of an object and its velocity. When the object moves with a velocity at which the arithmetic total of the object’s kinetic energy, its gravitational potential energy equates to zero. This means the object should possess greater kinetic energy than the gravitational potential energy to escape to infinity.

The easiest way to understand escape velocity formula derivation is by using the concept of conservation of energy. Let’s think that an object is trying to fly from a planet (that is uniform circular in nature) by going away from it.

The main force behind such an object will be the planet’s gravity. We already know that kinetic energy (K) and the gravitational potential energy (U

_{g}) are the only two kinds of energies related here.

So by following the principle of conservation of energy, we can write:

\[(K + U_{g})_{i} = (K + U_{g})_{f}\]

\[K = \frac{1}{2} mv^{2}\]

\[U = \frac{GMm}{r}\]

Here U_{gf} is considered zero as the distance is infinity and K_{f} will also be zero as final velocity will be zero.

The minimum velocity needed to escape from the gravitational force of the massive body is represented by:

\[V_{e} = \sqrt{2gr}\]

Where,

\[g = \frac{GM}{r^{2}}\]

The content below will help to derive an expression for escape velocity.

The acceleration due to gravity (earth), g = 9.8 m/s

^{2}.The radius (earth), R = 6.4 × 10

^{6}m.The escape velocity (earth), v

_{e}= √2 × 9.8 × 6.4 × 10^{6}.Therefore, v

_{e}= 11.2 × 10^{3}m/s = 11.186 km/s or 11.2 km/s (Approximately).

The escape speed of the earth at the surface is approximately 11.2 km/s. This means to escape from earth’s gravity and travel to infinite space, an object must have a minimum of 11.2 km/s of the initial velocity.

The unit of escape speed or escape velocity is expressed in meter per seconds (m.s^{-1}). This is also the SI unit of escape speed.

Dimensional formula of the earth’s mass = M1L0T0.

Dimensional formula of universal gravitational constant = M-1L3T-2.

Dimensional formula of the centre of the earth to the distance covered = M0L1T0.

So, after substituting in the equation the dimensional formula of escape speed is = M0L1T-1.

The table below shows the escape velocity of various objects:

There exists a relationship between escape velocity and orbital velocity. The relationship between the escape velocity and orbital velocity is proportional in nature. Escape velocity refers to the minimum velocity needed to overcome the gravitational pull of the massive to fly to the infinite space. Orbital velocity is a velocity that is required to rotate around a massive body. This means if the orbital velocity increases, the escape velocity also increases and if orbital velocity decreases, the escape velocity also decreases.

FAQ (Frequently Asked Questions)

Q1: What is Escape Velocity Derivation?

Ans: Escape velocity derivation is a very popular concept in the kinematics topics of physics. To derive an expression for escape velocity, it is important to understand all the concepts in-depth and needs to have a clear understanding of the related topics. It is easy to calculate the minimum velocity of an object requires to overcome a particular planet’s or object’s gravitational pull. Here, the derivation of escape velocity is outlined in a straightforward and easy to understand manner that will help to learn the concept without any hassles.

To derive an expression for escape velocity, the formula is written is as follows:

Ve = √2gR

Q2: What is the Escape Velocity Formula?

Ans: The massive bodies such as planets, stars which are spherically symmetrical, for any given distance, the escape velocity formula derivation is mathematically expressed as:

Ve = 2GMr

Ve represents the escape speed.

G represents universal gravitational constant. (G ≅ 6.67 × 10

^{-11}m^{3}kg^{-1}s^{-2}).M represents the mass of the massive body. (the body from which the object is going to escape).

r represents the distance from the centre of the massive body to the object.

This above-mentioned relation explains what is the formula of escape velocity. It is independent of the mass of the object which will be escaping the massive body.