Free Fall of Objects Considering the Acelleration of the Planet

1. Nov 25, 2013

Gilbfa

Dear members,

My name is Gilberto F. A. and I would like to ask your help regarding a topic in Gravitation. I have formal education in engineering and recently, as part of an attempt to refresh my knowledge of physics I started to study again on my spare time whenever I can.

A few days ago I came across a subtle problem regarding Gravitation which is puzzling me. Just to be clear this is not homework nor a prank. The question is the following:

Do bodies with different masses falling from the same height, with zero initial velocity really hit the ground of a planet at the same time when the acceleration of the planet is also considered?

Regarding this question I would like to ask your opinion and please read the entire message before answering as this question of mine is really subtle.

By the way I am aware of Galileo's experiment about the free falling objects from a tower and of the experiment made at the moon regarding the simultaneous falling of a feather and a hammer.

Before going into much detail I would like to state the hypothesizes of this “experiment”. Here they are:

- Consider there is no air resistance or any other dissipative resistance to motion;
- Consider that the planet where the experiment will be carried-out does not rotate around its axis, nor does this planet orbit another celestial body;
- Consider that the velocities involved in the experiment are much lower than the speed of light so that relativistic effects need not be considered;
- Consider two falling objects with different masses falling at this planet. The first with mass m1 (named “Body 1”) and the second with mass m2=2*m1 (named “Body 2”);
- Consider this is a “small planet” with mass m3=10*m1;
- Consider that each object will fall exactly from the same height such that the distance between the centers of the bodies involved in the experiment is “d” and that they fall with initial velocity equal to zero;
- Consider that each object will fall in different occasions and in isolation, that is:
Case 1: When the first body is falling consider that the only bodies present are m1 (“Body 1”) and the planet;
Case 2: When the second body is falling consider that the only bodies present are m2 (“Body 2”) and the planet;
- Consider also the acceleration of the “small planet” toward the falling body.

Considering the hypothesizes above I will examine Case 1 first and then Case 2.

Case 1:

Considering Newton's Gravitation Law for two bodies with masses m and m' separated by a distance “d”, F=G*m*m'/d^2, the acceleration of the first body (“Body 1”) towards the planet is a1= G*m3/d^2.

Doing the same for the acceleration of the planet towards the first body, you get ap= G*m1/d^2

Case 2:

In this case, the acceleration of the second body (“Body 2”) towards the planet is a2= G*m3/d^2, which is the same as the acceleration of “Body 1” towards the planet.

Regarding the acceleration of the planet towards the second body, its value is ap = G*m2/d^2 = 2*G*m1/d^2, which is twice the value of the acceleration of the planet towards “Body 1”.

Comparing these 2 cases, you see that the accelerations of the 2 bodies towards the planet are the same, however the accelerations of the planet towards each body are different. In this view, the body with a bigger mass will hit the planet's ground in a shorter period of time.

What am I missing here?

Best regards,

Gilberto F. A.

2. Nov 25, 2013

Bandersnatch

Hi Gilberto, welcome to PF!

Nothing, your reasoning is correct. The "all bodies fall equally fast" statement is true only for masses of falling bodies much lower than the mass of the planet, where motion of the planet can be neglected.

3. Nov 26, 2013

Gilbfa

Hi Bandersnatch,

Thank you for your reply! I guess I missed this subtle point the first time I studied this subject.....

Best regards,
Gilberto