BenjaminSTR
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The planet Melpomene has three moons, Othello, Hamlet and Romeo, each moving at a
constant speed in the same direction in circular orbits.
Melpomene
Othello
Hamlet
Romeo
Othello takes 20 days to complete one orbit of Melpomene, whereas Hamlet takes 45 days and
Romeo takes 120 days.
How often are Melpomene and all three of its moons in line with each other in the order shown
in the diagram?

A every 36 days
B every 72 days
C every 132 days
D every 180 days
E every 216 days

-The solution says that B is correct but WHY?-
Thank you
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astro67
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(Original post by BenjaminSTR)
The planet Melpomene has three moons, Othello, Hamlet and Romeo, each moving at a
constant speed in the same direction in circular orbits.
Melpomene
Othello
Hamlet
Romeo
Othello takes 20 days to complete one orbit of Melpomene, whereas Hamlet takes 45 days and
Romeo takes 120 days.
How often are Melpomene and all three of its moons in line with each other in the order shown
in the diagram?

A every 36 days
B every 72 days
C every 132 days
D every 180 days
E every 216 days

-The solution says that B is correct but WHY?-
Thank you
Imagine that the planets start off in alignment at time t = 0. One approach is to write expressions for the total full and partial revolutions that each planet will undergo in time t:

p1 = t/20
p2 = t/45
p3 = t/120

Planets line up again when p2-p1, p3-p2, and p3-p1 are all integer numbers of revolutions, i.e. all planets have moved the same amount round their orbits, give or take a whole number of revolutions.

p2-p1 = t/20 - t/45 = 25t/(20*45) = N (i.e. some integer) for alignment

=> t = N*(20*45)/25 = N*36

so Othello and Hamlet align every 36, 72, 108 days etc.

Similarly, you get a sequence of values for p3-p2 and for p3-p1. The smallest value of t these all have in common is your answer.
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