Acceleration in Gravitational Field

[philipp2020]

Hi

It is always said that in a vacuum two objects even if they have different mass, have the same acceleration to a gravitational point regardless of their mass.

If I understand this right, it needs the same time for 1kg mass or 1000kg to fall
from 100m to the gravitational center.

But what I don't understand is, that the 1000kg mass itself has a much stronger gravitational field so that it also accelerates the gravitational center. So all together the time is shorter for the 1000kg mass and the gravitational field to clash than it is for the 1kg mass.

 

[SHISHKABOB]

the net force on an object is given by F = ma

the force due to gravity between two objects is equal to $F = G\frac{Mm}{r^{2}}$

setting these two equal to each other we get

$ma = G\frac{Mm}{r^{2}}$

dividing through by m gets

$a = G\frac{M}{r^{2}}$

which shows that the acceleration on an object due to the force of gravity acting on it by another object is independent of the mass of the other object

in $F = G\frac{Mm}{r^{2}}$ it already takes into account the gravitational field of the "test mass", but when we solve for the acceleration of that test mass, its own mass does not matterall of this assumes that the mass M is held stationary. If it is not held stationary, then you are correct: the time taken for the larger mass to accelerate towards the mass M would be less than that of the smaller mass.

 

[philipp2020]

thanks very much! I didn't know about the stationary assumption.

 

[Buckleymanor]

all of this assumes that the mass M is held stationary. If it is not held stationary, then you are correct: the time taken for the larger mass to accelerate towards the mass M would be less than that of the smaller mass.

Are you sure that is allways the case, won't the acceleration be proportional.
Thats why all objects on Earth fall at the same rate.
Take the Earth and remove a 1Kilo mass from it and drop it from a height.
The 1 kilo mass will accelerate towards the Earth at a far greater speed than the Earth accelerating towards the 1 Kilo mass.
Like wise if you split the Earth in two each half would accelerate towards each other at a total speed equivalent to the 1 Kilo mass and the Earth's acceleration towards the 1Kilo mass.
It don't have to be held stationary what is required to make a difference is the size of the gravitational centre.
Two 1 Kilo masses in vacue are not going to accelerate towards there gravitational centre at the same speed as a 1000 Kilo mass and a 1Kilo mass.

 

[philipp2020]

ok let's make an easy example

lets put a second earth1 beside earth. Earth accelerates Earth1 with a1 and likewise Earth1 accelerates Earth with a2.

Then let's put a spacehuttle there. Earth accelerates the spaceshuttle with a3 and likewise the spaceshuttle accelerates Earth marginally with a4.

Then in my understanding a1 = a2 = a3 > a4.

So the time until they collide in a 2 vacuum space with two obects is shorter in the first case.

 

[Doc Al]

all of this assumes that the mass M is held stationary. If it is not held stationary, then you are correct: the time taken for the larger mass to accelerate towards the mass M would be less than that of the smaller mass.

I would say that it gives the acceleration of the mass with respect to an inertial frame. To find the relative acceleration between the masses, you'd have to consider the acceleration of both masses.

Of course if one mass is monstrously larger than the other, such as the Earth is compared to a 1000 kg mass, the acceleration of the larger mass can often be neglected.