# Is body to body gravity scale independent?

• dylankarr.com
In summary: Overall, the summary is that smaller and closer bodies fall into each other faster due to their lower mass and shorter distance between them.
dylankarr.com
I'm working on a gravity simulator and I can't help but feel that I've done something wrong. Extremely large bodies take a longer time to fall into each other than smaller ones. Is this right? Like for example, if you had to bodies of solar mass at 1 AU apart, would it take a longer, shorter or the same amount of time for two other bodies of half solar mass at half an AU to fall into each other?

dylankarr.com said:
if you had to bodies of solar mass at 1 AU apart, would it take a longer, shorter or the same amount of time for two other bodies of half solar mass at half an AU to fall into each other?

In the second case (two 0.5-solar mass bodies), the force is 1/4 as much, the acceleration is 1/2 as much, and the time is greater by a factor of $\sqrt{2}$.

dylankarr.com said:
I'm working on a gravity simulator and I can't help but feel that I've done something wrong. Extremely large bodies take a longer time to fall into each other than smaller ones. Is this right? Like for example, if you had to bodies of solar mass at 1 AU apart, would it take a longer, shorter or the same amount of time for two other bodies of half solar mass at half an AU to fall into each other?

In both cases the force is the same, $$F = \frac{G M_1M_2}{r^2} = \frac{G (M_1/2) (M_2/2)}{(r/2)^2}$$

The acceleration is different. In the first case:

$$a = \frac{GM_1}{r^2}$$

In the second case:

$$a = \frac{G(M_1/2)}{(r/2)^2} = \frac{2GM_1}{r^2}$$

so the acceleration is twice that of the first case.

The distance (x) to travel is also half that of the first case, so:

In the first case $$t= \sqrt{\frac{2x}{a}}$$

for the time to fall a short distance x relative a large r.

In the second case $$t = \sqrt{\frac{2(x/2)}{2a}} = \sqrt{\frac{x}{2a}}$$

which is less time to fall in the second case.

You would of course have to integrate the equations over the total distance for them to come together and the analysis above is just for a short segment of the fall, but overall the smaller closer bodies fall together faster.

If the initial distance apart (r) was the same for the smaller and larger bodies, then the falling time would for the lighter bodies would be longer because:

For the larger bodies $$t = r\sqrt{\frac{8x}{GM_2}}$$

and for the smaller bodies $$t = r\sqrt{\frac{16x}{GM_2}}$$

Last edited:

## 1. What is body to body gravity scale?

Body to body gravity scale is a type of measurement used in physics to determine the gravitational force between two objects. It involves measuring the distance and mass of two objects and using Newton's law of gravitation to calculate the force between them.

## 2. How does body to body gravity scale work?

Body to body gravity scale works by using a balance mechanism to measure the distance and mass of two objects. The distance is measured using a ruler or other measuring device, and the mass is usually determined by using a scale. These values are then plugged into the equation for Newton's law of gravitation to calculate the force between the two objects.

## 3. Is body to body gravity scale independent of other factors?

Yes, body to body gravity scale is independent of other factors such as air resistance, friction, and the presence of other objects. This is because the equation for Newton's law of gravitation only takes into account the distance and mass of the objects, not any external factors.

## 4. Can body to body gravity scale be used to measure the force between any two objects?

Yes, body to body gravity scale can be used to measure the force between any two objects, as long as the objects have mass and are at a measurable distance from each other. However, it may not be the most accurate method for measuring the force between very large or very small objects.

## 5. How accurate is body to body gravity scale?

Body to body gravity scale can be quite accurate if the measurements of distance and mass are precise. However, there may be some limitations in its accuracy when measuring the force between very large or very small objects, or in situations where external factors cannot be completely eliminated.

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