Class 11 > Unit # 05: Work, Energy & Power > Escape Velocity

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Gravitation & Work - Escape Velocity

Q. Define escape velocity and derive expression for escape velocity.

Definition

"Escape velocity on Earth or any other planet is defined as the minimum initial velocity with which a body must be projected vertically upwards from the surface so that it just crosses the gravitational field of the planet and never returns on its own."

The required work to escape the gravitational field is done entirely at the cost of the kinetic energy imparted to the body at the surface of the planet.

Figure: Projection of a body from the surface of Earth to overcome gravitational attraction.

Derivation of Escape Velocity

Let $v_{es}$ be the escape velocity of a body of mass $m$, and let $M_e$ and $R_e$ be the mass and radius of the Earth, respectively. For the body to completely escape the Earth's gravitational field, its initial kinetic energy at the surface must equal the magnitude of its gravitational potential energy (absolute potential energy) at the surface.

$\text{Kinetic Energy} = \text{Potential Energy}$

$\frac{1}{2} m v_{es}^2 = \frac{G M_e m}{R_e}$

Canceling the mass $m$ of the body from both sides:

$\frac{1}{2} v_{es}^2 = \frac{G M_e}{R_e}$

$v_{es}^2 = \frac{2 G M_e}{R_e}$

Taking the square root on both sides:

$v_{es} = \sqrt{\frac{2 G M_e}{R_e}} \quad \text{--- (i)}$

Expressing Escape Velocity in terms of Gravitational Acceleration ($g$)

Since the gravitational acceleration at the Earth's surface is given by:

$g = \frac{G M_e}{R_e^2} \implies G M_e = g R_e^2$

Substituting $G M_e = g R_e^2$ into equation (i):

$v_{es} = \sqrt{\frac{2 (g R_e^2)}{R_e}} = \sqrt{2 g R_e}$

Numerical Value

Substituting the standard values for Earth ($g \approx 9.8 \, \text{m/s}^2$ and $R_e \approx 6.4 \times 10^6 \, \text{m}$), the escape velocity comes out to be approximately: $$v_{es} \approx 11.2 \, \text{km/s}$$

Note: The value of escape velocity depends directly on the mass and radius of the planet from whose surface the body is projected. Hence, the escape velocity is different for different planets and celestial bodies.

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