Question

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?

Answer 1

`|v| =√( G \cdot M / r)`, where

• 
  `M` the mass of the planet, and
• 
  `G` the universal gravitation constant.

Step-by-step 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,
`\Sigma F` to its orbit velocity
`v` and its mass
`m` required for it to stay in orbit :

`\Sigma F = m \cdot v^(2) / r` (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
`\Sigma F` acting on the soccer ball shall equal to its weight,
`W = m \cdot g` where
`g` the gravitational acceleration constant. Thus

`\Sigma F = W = m \cdot g` (equation 2)

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

`m \cdot g = \Sigma F = m \cdot v^(2) / r \\ v^(2) = g \cdot r \\ |v| = √(g \cdot r) \; (|v| \ge 0)` (equation 3)

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

`g = G \cdot M/ r^(2)` (equation 4)

where the universal gravitation constant
`G = 6.67408 * 10^(-11) \cdot \text{m}^(3) \cdot \text{kg}^(-1) \cdot \text{s}^(-2)`

Thus

`\begin{array}{lll}|v| &=& √(g \cdot r)\\ & =&√( G \cdot M / r)\end{array}`