Physics puzzle ratios of two times which it take an object to fall down.

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nickthrop101
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Homework Statement


What is the ratio between times for the aliens to hit the floor. you have two spheres of equal density throughout. One has a radius R and the other 1/2R. An alien falls down one and takes time T, the other it falls down in time t. In the second on, time t, half of the mass of the ball is taken out and it only haas to fall down half of r to the center. What is the ratio between times?


Homework Equations


i think 4(pie)r^2
v=v+ta^2
(v+ta^2)T=d

The Attempt at a Solution


if the mass was the same, it would take a time ^2 between them as acceleration is squared due to gravity, so i think it would just be Ta^2:1/2T^2
i know this is wrong but I am not sure how to go on
 
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I'm puzzled by the wording. Could you provide the exact wording of the question and any diagram that comes with it?
An alien falls down one
One what? If one of the spheres, are we sliding down the outside or falling inside of the the sphere? Are the aliens falling near the surface of the Earth or what? The 1/(2R) sphere will likely be quite small, so the size of the alien will probably be important.
 
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sorry i was typing this quite quickly, the question was given to me by my physsics teacher to remember so no diagram but it says thaat there is a mining project on an alien planet to the center of the planet and the alien hyperthetically falls down the mineshaft to the bottom of the shaft (The centre of a planet) Then in asimilar situation but half the radius and half the mass of the first sphere an alien falls into the center of it :D is that okay
 
This is a very different question from the first post!
"Half the radius and half the mass" is very surprising since mass is 4/3*pi*R³*Density - with 1/2 the radius you would expect 1/8 the mass. Unless the density is much greater for the second planet.

This problem is actually quite difficult since the force of gravity is GMm/R² where M is the mass of the planet up to radius R. So the force varies with R during the fall. Acceleration is not constant, and you can't use constant acceleration formulas. You will need to use calculus or a digital (spreadsheet?) model to work out the time to fall.