A baseball pitcher can throw a fastball at a speed of 150 km/hr. What is the largest size spherical asteroid of density rho=3 g/cm^3 from which he can throw the ball fast enough that it:
(a) escapes from the asteroid into heliocentric orbit?
(b) rises to a height of 50 km?
(c) goes into a stable orbit about the asteroid?
KE = 1/2mv^2 (kinetic energy)
PE = GmM/r (potential energy, gravitational)
Fc = mv^2/r (centrifugal force)
Fg = GmM/r^2 (gravitational force)
The Attempt at a Solution
v = 150 km/hr ~ 42 m/s
rho = 3 g/cm^3 = 3000 kg/m^3
M = rho*V
V = 4/3*pi*r^3
(a) make KE = PE and solve for r, which i get r = v*sqrt(3/8*pi*rho*G) = 32439 m
(b) not sure...
(c) make Fc = Fg and solve for r, which i get r = v*sqrt(23/4*pi*rho*G) = 45876 m
Did I do (a) and (c) correctly, and how should I approach (b)? Many thanks!