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Alien Messages (Posted on 2007-01-23)

In another solar system there exists a planet approximately the size of Earth. This planet is a solid sphere of uniform density.

The inhabitants send messages to each other by placing the message in a capsule and placing the capsule in a message tube. These message tubes are straight lines through the planet from one point on the surface to another.

A capsule moves through a tube only under the influence of gravity (no friction force between the capsule and the tube and no forces due to the planet's rotation).

What is the ratio between the time it takes to send a message from the North Pole to the South Pole and the time it takes to send a message from the South Pole to the Equator?

Note: The gravitational effects of the message tubes are negligible compared to the gravitational effects of the planet.

Note: Assume a mechanism at each end of a message tube to catch an incoming capsule (easy to do since the velocity will be zero).

Submitted by Bractals

Rating: 2.8000 (5 votes)

Solution: (Hide)

Let P and R be the endpoints of a message tube and Q its midpoint. Let O be the planet's center and C the position of the capsule in PR. Let s = |CQ| and r = |CO|. The force F_s on the capsule in the direction CQ is
F_s = (s/r)F_r
where F_r is the force (due to gravity) on the capsule in the direction CO
F_r = GMm/r²
where G is the gravitational constant, M is the mass of the solid sphere with center O and radius r, and m is the mass of the capsule. The mass M is given by
M = k(4πr³/3)
where k is the density of the planet. The equation of motion for the capsule is
-F_s = m d²s/dt² or -(s/r)Gk(4πr³/3)m/r² = m d²s/dt² or d²s/dt² + Ks = 0
where K = 4πGk/3 a constant. Solving this d.e. gives
s = (|PR|/2) cos(t√K)
The capsule starts at one endpoint when t√K=0 and reaches the other endpoint when t√K=π. The travel time for the message is π/√K. A constant that is independent of the length |PQ|. Thus, the ratio asked for is 1:1.

Note: The travel time for a message on a planet the size of the Earth and the same mass is approximately 42 minutes.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
re(3): from general knowledge (spoiler)	Charlie	2007-01-24 16:19:51
re(2): from general knowledge (spoiler)	Leming	2007-01-24 13:52:05
re: from general knowledge (spoiler)	Jer	2007-01-24 11:51:48
from general knowledge (spoiler)	Charlie	2007-01-24 10:20:21

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