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.
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.