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Moon Experiment (Posted on 2004-11-22) Difficulty: 5 of 5
Assume the moon is a perfect sphere and a straight tunnel has been drilled through the center. How long would it take a 1kg ball dropped from one end of the tunnel to reach the center? Ignore all resistances.

If a second 1kg ball is dropped 10 seconds after the first one, when and where in the tunnel would they first meet?

Idealized Moon Stats:
- Diameter: 3480 kilometers
- Mass: 7.38x10^22 kilograms (uniform density)

See The Solution Submitted by Brian Smith    
Rating: 2.4000 (5 votes)

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part 1: another method; another answer | Comment 5 of 22 |
(In reply to part 1: different answer by Charlie)

The following method is based on a surface acceleration to gravity of 1.6266  m/s^2  (from GM/r^2), and uses numerical integration of the velocity and distance traveled.  Acceleration due to gravity is assumed to be proportional to the cube of the distance from the center of the moon.  The time taken to reach the moon's center calculated by this method is 1917.6 seconds or 31.96 minutes.  An integration interval of 1/16 second was used.

The program is

DEFDBL A-Z
acc0 = 1.6266 ' m/s
intvl = .0625
r = 1
dist = 0    ' meters fallen
vel = 0

DO
  r = 1 - dist / (3480000 / 2)
  acc = acc0 * r * r * r
  newVel = vel + acc * intvl
  newDist = dist + intvl * (vel + newVel) / 2
  dist = newDist
  vel = newVel
  t = t + intvl
  IF t = INT(t) AND t MOD 100 = 0 THEN
    PRINT USING "##### ####.### ########.##"; t; vel; dist
  END IF
LOOP UNTIL dist > 3480000 / 2

PRINT USING "#####.## ####.### ########.##"; t; vel; dist

PRINT t / 60

The resulting table of speeds and depths at intervals of 100 seconds and at the end is:

  100  161.904     8114.07
  200  319.341    32231.44
  300  468.210    71695.94
  400  605.085   125472.72
  500  727.435   192227.72
  600  833.726   270423.33
  700  923.413   358418.49
  800  996.832   454562.88
  900 1055.037   557277.25
 1000 1099.597   665115.17
 1100 1132.399   776804.64
 1200 1155.468   891270.63
 1300 1170.815  1007641.01
 1400 1180.327  1125239.39
 1500 1185.685  1243568.29
 1600 1188.312  1362285.73
 1700 1189.343  1481177.83
 1800 1189.609  1600129.13
 1900 1189.634  1719091.94
 1917.63 1189.634  1740059.24
 31.96041666666667

  Posted by Charlie on 2004-11-22 22:55:03
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