Question

Using the formula U = − Gm1m2 r for the gravitational potential object between two objects, calculate the velocity needed to escape an object’s gravitational field. That is to say that a rocket has enough initial velocity v at the object’s surface so that its kinetic energy completely cancels out the gravitational potential energy

Answer 1

Taking the case of earth, consider the mass of the rocket as 'm' and mass of the earth as M.

R is the radius of the earth.

Suppose the initial velocity of the rocket is 'v' m/s

So, kinetic energy of the rocket, KE = (1/2)*m*v^2

Potential energy of the rocket at the surface of the earth, U = G*M*m / R

[Please note that we have not consider the negative sign just to evaluate the value of velocity]

So, when the kinetic energy cancels the gravitational potential energy -

We have -

KE = U

=> (1/2)*m*v^2 = G*M*m / R

=> v^2 = (2*G*M) / R

Put the values of G, M and R -

v = sqrt[(2 x 6.674 x 10^-11x 5.974 x 10^24) / (6371 x 10^3)] = sqrt[0.01252 x 10^10] = 0.112 x 10^5 m/s = 11.2 km/s

So, the required velocity of the object = 11.2 km/s (Answer)