Law of universal gravitation - why DON'T heavy objects fall faster in a vacume

[Yay]

I'm trying to understand the basics of gravity. Newtons law of universal gravitation gives us : F1 = F2 = G (m1m2)/r2

Now on Earth you can simply that down to
F = mg where g = G (mearth/R2)

But using f = mg, a heavier object say a tennis ball should fall faster than a cannon ball. But it doesn't because for some reason acceleration due to gravity on Earth is about 9.857 ms, regardless of how heavy your object is.

This doesn't make much mathematical sense to me since the force of gravity is F = mg where m is the mass off the falling object, so a heavier object should fall quite faster !

Can someone please show me the step I'm missing here?

 

[diazona]

You're missing the step that tells you how fast something should fall. F is just the force on an object; it doesn't tell you anything how fast the object falls. For that you'd need to figure out the acceleration, and from it the velocity. And of course, Newton's second law says that acceleration is related to force by
$$F = ma$$

 

[espen180]

$$F=mg$$ Gravitation force

$$F=ma$$ Newton's 2nd law

Combined becomes:

$$a=g$$

An acceleration not dependent of mass.

The same holds for the universal law. The acceleration only depends on the mass of the other object, not the one being accelerated. (Thogh both are accelerated).

 

[Pengwuino]

Remember, the force isn't the acceleration, it simply causes the acceleration.

 

[YupHio]

The amount of inertia in a heavier object resists it from accelerating faster. That's how I think of it (just made that up).

 

[Yay]

Thanks guys, I understand it now :)

 

[jrab227]

The amount of inertia in a heavier object resists it from accelerating faster. That's how I think of it (just made that up).

I like this one.

I personally like to think of it as sand, and I assume upper level physicists construct the graviton from this idea.

Basically, if we assumed somewhere in the sub-atomic particles was 1 "graviton", then each graviton would be attracted to, say, Earth's gravitons, in exactly the same way. I tend to think of this as sand falling to the earth. If one throws up sand (assuming all of the grains are the same shape and mass, although I'll explain why mass doesn't matter), they all fall down at the same rate. So let's say I have an "object" with sand inside of it, and a smaller volume "object" with sand inside of it, they should fall at the same rate, since each grain of sand will fall at the same rate. Basically it doesn't matter what the orientation of a pile of sand is or how many other grains are near the sand, (or it doesn't matter if the sand is stuck together), every grain will fall at the same rate. Applying this to real matter, we assume at a base particle level, somewhere, there is a "sand" particle(the graviton). Let's say for example an electron, a proton, and a neutron each had 1 grain of "sand" each, then each would fall at the same rate, in theory, holding the atom together. In this way, it doesn't matter if the atom were hydrogen, or something heavier, and it wouldn't matter if the atom was part of an elephant, or a baby. Although a graviton has yet to have been found, the theory of the graviton holds together because all objects fall at the same rate, allowing the idea that gravity effects small bits rather than objects as a whole.

 

[rcgldr]

The rate of closer of heavier objects is faster than the rate of closer of lighter objects, but the rate of acceleration of one object is proportional to the mass of the second object:

F = G m₁ m₂ / r²

a₁ = F / m₁ = G m₁ m₂ / (r² m₁ ) = G m₂ / r²

a₂ = G m₁ / r²

However if you're standing on m₂, then it will appear that m₁ falls faster by a₂ depending on how massive m₁ is.

 

[DocZaius]

Well technically heavier objects do "fall" faster in a vacuum. But you must keep two things in mind when saying that.

First you must take the law of universal gravitation (F= Gm1m2/r^2) into account instead of the simplified F = mg version. Then you must define the act of falling as an act that culminates in the meeting of the two objects and not simply look at the acceleration of one of the two objects involved, which is what we do now in regards to an object falling towards earth.

Having kept those things in mind, m1 and m2 will meet later than m1+x and m2 will meet (where x is a positive amount of mass), assuming same starting conditions.

The reason for this is that although both m1 and m1+x will accelerate just as much in each scenario, m2 will accelerate more in the second scenario.

Also note that if we defined the act of falling as only the acceleration that m1 or m1+x experience, then m1+x does not fall faster.

 

[rcgldr]

In case you're curious, the math for the time it takes for objects at rest in space to collide involves solving an intergral figured out by arildno and is explained in these two threads. The first thread shows more of the intermediate steps, but assumes point masses, the second case assumes finite sized and equal masses.

https://www.physicsforums.com/showthread.php?t=360987

https://www.physicsforums.com/showthread.php?t=306442