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## Homework Statement

A space ship 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)