Question

Define the term orbital speed. Establish a relation for orbital speed of the earth. Find the ratio of this orbital speed and escape speed

Answer 1

The speed at which a celestial body revolves in its orbit is called orbital speed.
Suppose, ‘v’ is the orbital speed of earth. Then the centripetal force on earth must be provided by the gravitational force between earth and sun. So,
mv²/d = GMm/d²
Where, m is mass of earth, d is distance between center of earth and sun, M is mass of sun.
So,
v = (GM/d)^1/2
Escape velocity of earth is given by, v_e = (2gR)^1/2
[R is the radius of earth]
=> v_e = {2(GM/R²)R}^1/2
=> v_e = (2GM/R)^1/2
So,
v/v_e = [(GM/d)^1/2]/[ (2GM/R)^1/2]
=> v/v_e = (R/2d)^1/2