Calculating impact velocity of freefalling object with increasing grav

[TheBladeRoden]

Homework Statement

A spaceship is in a vertical free fall towards an airless moon. It starts out at 1,247,000m above sea level and has an initial vertical speed of -132 m/s. The moon's radius is 200,000m and gravity at sea level is 1.63 M/s^2, but decreases with distance. Calculate the vertical velocity the ship will have by the time it impacts the moon.

Homework Equations

t=(vf-vi)/a

t=time
vf = velocity final
vi = velocity initial
a = acceleration


d=(1/2)at^2

d = distance


g1=g0/((r1/r0)^2)

g1 = initial altitude gravity
g0 = sea level gravity
r1 = initial altitude radius
r0 = sea level radius

The Attempt at a Solution

t0 = 0
v0 = 132
h0 = 1217000
a0 = 0.0311

t1 = t0+1
v1 = v0+a0
h1 = h0-v0
a1 = g0/(v1/r0)^2

final velocity with constant lowest gravity

vg1=278.68m/s=SQRT(2*g1*(r1-r0))


final velocity with constant sea level gravity

vg2=2016.24m/s=SQRT(2*g0*(r1-r0))


average the two results = 1147.46m/s=(vg1+vg2)/2

square average the results = 2035.41m/s=SQRT(vg1^2+vg2^2)

 

[gneill]

Averaging results won't work here with acceleration varying with the inverse square of the distance.

A better approach might be energy conservation.