Escape Speed

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

What is the Escape Speed?

Escape speed is defined as the minimum speed with which a mass needs to be propelled from the earth’s surface to escape the 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 be free from the gravitation force of a massive object.

What Speed is Escape Velocity?

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.

Derivation of Escape Speed

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 gravitational potential energy (Ug) 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

Where,

\[K = \frac{1}{2}mv^{2}\], the kinetic energy

\[U = \frac{GMm}{r}\], the gravitational potential energy

Here Ugf is considered zero as the distance is infinity and Kf 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:

\[V_{e} = \sqrt{2gr} = \sqrt{\frac{2GM}{r}}\]

Where

\[g = \frac{GM}{r^{2}}\], G is the universal gravitational constant.

What is the Value of Escape Velocity of the Earth?

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

• The acceleration due to gravity (earth), g = 9.8 m/s2.

• The radius (earth), R = 6.4 × 106 m.

• The escape velocity (earth), ve = 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. 

Escape Velocity refers to the minimum speed that is required in order to escape from a planet’s gravitational pull.

Taking an example, if we consider the earth to be a massive body, then the escape velocity is the minimum velocity which an object will have to acquire in order to be able to overcome the gravitational field of the earth and fly to infinity without falling back.

This will depend on the distance of the object from the massive body and the mass of the massive body. If the mass is more, it will be higher. Similarly, the closer the distance, the escape velocity will be higher.

What is the Unit of Escape Speed?

The unit of escape speed or escape velocity is expressed in meters per second (m.s-1). This is also the SI unit of escape speed.

Dimensional Formula

The dimensional formula of the escape velocity can be obtained by resolving the formula dimensionally. Now, we know that the escape velocity formula includes a constant, G, which is known as the universal gravitational constant. The value of the universal gravitational constant is  6.673 × 10-11 N . m2 / kg2. The unit for escape velocity is meters per second (m/s). 

Since the escape velocity is again a form of velocity and it is measured in terms of m/s. Thus the dimensional formula of escape velocity is [M0L1T-1].

We can also determine the dimensions of escape velocity by dimensional analysis. We know that the dimensional formula of the escape velocity is:

\[V_{e} = \sqrt{2gr} = \sqrt{\frac{2GM}{r}}\]

Now, we know that,

Dimensional formula of the earth’s mass =M= M1L0T0.

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

Dimensional formula of the center of the earth to the distance covered =r= M0L1T0.

The dimensional formula of escape speed is Ve= M0L1T-1

Therefore,

\[\sqrt{\frac{2GM}{r}} = \left ( \frac{[M^{-1}L^{3}T^{-2}][M]}{[L]} \right )^{\frac{1}{2}} = [M^{0}L^{2}T^{-2}]^{\frac{1}{2}} = [M^{0}L^{1}T^{-1}] \]

Therefore, the dimensional formula of escape velocity is \[[M^{0}L^{1}T^{-1}] \]

The table below shows the escape velocity of various objects:

The Escape Velocity of Various Objects

Did You Know?

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.