Planets are big.

An asteroid has a radius of 580 km and a gravitational acceleration at the surface of 4.0 m/s2.

How far from the surface will a particle go if it leaves the asteroid's surface with a radial speed of 1000 m/s?  (It never escapes orbit)

>>6

unless I'm missing something entirely... the mass of the asteroid M isn't specified
you're going to need to calculate the energy required to lift the particle from the surface, r (radius of the asteroid) to some final distance, Rf (final radius)...

the energy required is = int(g*m*(r/s)^2)ds from Ra to Rf...
s = distance from the center of the object
everything is constant except s, so this equals:

mgr^2[-Rf^(-1) - r^-1 ] = mgr - m*g*r^2/Rf

take this equal to the object's intial kinetic energy, 1/2*m*v^2

i.e. 1/2*m*v^2 = m*g*r - m*g*r^2/Rf, the mass cancels out leaving

1/2v^2 = g*r - g*r^2/Rf

solving for Rf, Rf = g*r^2/(g*r-1/2*v^2)

which makes since, because if the intial velocity is zero, Rf is just r, the radius of the asteroid (you didnt go anywhere)

plugging in g=4m/s^2, r=580*10^3m, v = 1000m/s


Rf = 740*10^3m (740 km) (2 sig figs).

anyone else verify this?