ICSE Class 10 Answered

What is the Escape Velocity?

Asked by preranapatil4360 | 25 Dec, 2021, 07:55: PM

Expert Answer

Escape velocity is defined as minimum velocity required for a particle at the surface of planet

so that the particle reaches a far point away from planet where the gravitational force of attraction

acting on the particle is zero .

Let us consider the escape velocity requirement of a particle on earth's surface

On surface of earth , let us consider the particle is projected with speed v upwards.

Total energy on surface of earth = Kinetic energy + Potential energy

Total energy on surface of earth = (1/2) m v² - ( G M m ) / R ....................................(1)

where m is mass of particle. Second term in above expression is potential energy ,

where G is universal gravitational constant ,M is mass of earth and R is radius of earth .

At far away point ( at infinity ) , potential energy is zero and let us consider the speed is negligible

compare to initial speed at earth's surface

Hence at infinity , total energy is zero

Hence by conservation of energy , (1/2) m v² - ( G M m ) / R = 0

After simplification , above expression becomes

begin mathsize 14px style v space equals space square root of 2 space G M over R space end root space equals space square root of 2 space open parentheses fraction numerator G M over denominator R squared end fraction close parentheses R end root space equals space square root of 2 space g space R end root end style

Where g = G M / R² , is the acceleration due to gravity on earth surface

if we substitute R = 6378 km in above expression , we get

begin mathsize 14px style v space equals space square root of 2 space g space R end root space space equals space square root of 2 cross times 9.8 cross times 6378 cross times 1000 end root space space equals space 1.118 space cross times space 10 to the power of 4 space m divided by s end style

Hence escape velocity on earth's surface = 11.18 km/s

Answered by Thiyagarajan K | 25 Dec, 2021, 09:54: PM

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