Suppose you are standing on top of a hemisphere of radius r and you kick a soccer ball horizontally such that it has velocity v. what is the minimum v that ensures the ball will not hit the hemispherical surface. in this case, how far does the ball land from the base of the hemisphere?

, where

Explanation:

Minimizing the initial velocity of the soccer ball would minimize the amount of mechanical energy it has. It shall maintain a minimal gravitational potential possible at all time. It should therefore stay to the ground as close as possible. An elliptical trajectory would thus be unfavorable; the ball shall maintain a uniform circular motion as it orbits the planet.

Equation 1  (see below) relates net force the object experiences, to its orbit velocity and its mass required for it to stay in orbit :

(equation 1)

The soccer ball shall experiences a combination of gravitational pull and air resistance (if any) as it orbits the planet. Assuming negligible air resistance, the net force acting on the soccer ball shall equal to its weight, where the gravitational acceleration constant. Thus

(equation 2)

Substitute equation 2 to the left hand side of equation 1 and solve for ; note how the mass of the soccer ball, , cancels out:

(equation 3)

Equation 4 gives the value of gravitational acceleration, , a point of negligible mass experiences at a distance from a planet of mass (assuming no other stellar object were present)

(equation 4)

where the universal gravitation constant

Thus

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