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

[nickthrop101]

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 betwen 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

 

[Delphi51]

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.

 

[nickthrop101]

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

 

[Delphi51]

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.

 

[asteriatic]

about it.

I would approach this problem by first identifying the relevant equations and principles that apply to the situation. In this case, we are dealing with the concept of acceleration due to gravity and the equations of motion. The equation for acceleration due to gravity is a = g, where g is the acceleration due to gravity (9.8 m/s^2 on Earth). The equations of motion are v = u + at and s = ut + 1/2at^2, where v is final velocity, u is initial velocity, a is acceleration, t is time, and s is displacement.

Next, I would analyze the given information and try to understand the scenario. We have two spheres of equal density, but one has a radius of R and the other has a radius of 1/2R. An alien falls down one sphere in time T, and the other sphere has half of its mass removed and the alien falls down in time t. The second sphere also only has to fall half of the radius to reach the center.

Based on this information, I would first calculate the acceleration due to gravity for both spheres using the equation a = g. Since the spheres have equal density, their masses would also be equal. The mass of the first sphere would be 4/3 πR^3ρ, where ρ is the density of the sphere. The mass of the second sphere would be 4/3 π(1/2R)^3ρ = 1/8(4/3 πR^3ρ) = 1/8m, where m is the mass of the first sphere. Therefore, the acceleration due to gravity for the first sphere would be g = GM/R^2, where G is the gravitational constant, and for the second sphere it would be g' = GM/2R^2.

Next, I would use the equations of motion to calculate the time taken for each sphere to fall. For the first sphere, we have s = R, u = 0, a = g, and t = T. Plugging these values into the equation s = ut + 1/2at^2, we get R = 1/2gT^2, or T = √(2R/g). For the second sphere, we have s = 1/2R, u = 0, a = g', and t = t. Plugging these values into