How can I prove that all objects, regardless of mass fall with the same acceleration?

Question

I know Newton's 2nd law Fnet=ma, and yet I have no idea how to prove this, I think the question is asking for some invented math problem, and this troubles me greatly.

I am solving a home work. I am not looking to cheat, I am looking for some loving guidance, because I do not know how to approach this question, but I have to prove the point without dropping any objects, or in other words, mathematically, I think.

Answer 1

You start dropping a feather and a hammer in a vacuum and see which one hits the ground first. You can't get much more convincing than empirical evidence. If what your teacher is asking for is to show that all objects having the same acceleration due to gravity regardless of mass follows from Newton's laws (which is not the same as proving it true, since you first have to demonstrate that Newton's laws are true, and that can only be done empirically), you might do something like this:

Let F be the force due to gravity, m be the mass of the object, M be the mass of the object attracting it, r be the distance between the two objects, and G be the universal gravitational constant. Then Newton's law of universal graviation states:

F=GmM/r²

However F=ma by newton's second law, thus:

ma=GmM/r²

Cancel the m:

a=GM/r²

As you can see, nowhere in the equation for the acceleration due to gravity does the mass of the object appear. It therefore follows that the acceleration due to gravity is independent of an object's mass. Q.E.D.

Answer 2

Yes, F = ma, but when a becomes g, the acceleration due to the pull of gravity, F = ma becomes W = mg; where the force is now called weight by convention. As g is a constant due to gravity W has to vary with mass; so that (W/m) = g the constant for all mass.

The simplest experiment is to take two balls the same size, but one is filled with something; so it is heavier than the hollow one. So W1/m1 = g and W2/m2 = g for ball 1 and ball 2. Since g is the same for both balls, they will accelerate at the same constant rate when dropped from a height h. h = 1/2 a t^2 = 1/2 g t^2; where h is the height the two balls are simultaneously dropped from.

Because they are dropped from the same height, we can write 1/2 g t1^2 = 1/2 g t2^2; so that, by cancelling like factors, we have t1^2 = t2^2 or t1 = t2. That is, no matter what the weight and mass of the two balls, they will hit the ground at the same time when dropped from the same height.

Make sure you use balls of the same size. If you do not, the larger ball will have a larger drag force on it than the smaller one. In which case, the larger ball will have an upward force on it greater than the small ball. In which case, the smaller ball will fall faster and the experiment will fail.

Case in point: a large balloon filled with air will fall slower than a tennis ball filled with air even if they weigh the same. That's because the drag force on the larger balloon will be greater than that acting on the smaller tennis ball.

PS: The previous answerer's proof is OK, but (W/m) = g = constant also proves the case mathmatically. In fact, g = GM/r^2, the first answerer's solution. (W/m) = g says that no matter what the mass or weight of an object, it will fall at the same rate of acceleration. So bodies dropped from the same height will hit the ground concurrently.

Answer 3

If I'm not wrong, the formula to prove this is another one, if you talk about proving the acceleration of gravity. Take a look at Newton's Universal Law of Gravitation, maybe you'll find the solution to your problem there.

Answer 4

There is a classic experiment that you could reproduce at home if you had access to a vacuum chamber. You take a long skinny vacuum chamber with a coin and a feather. Remove the air from it, and the feather, which no longer has any air resistance on it, though it is much lighter than the coin, it will fall at the same rate as the coin.

Answer 5

Do you know the acceleration due to gravity? 9.8 m/sec. Regardless of the mass, the accelleration due to gravity is always constant. The gravitational field of the earth does not change. Now, of course, the force with which an object strikes the earth will change as mass changes. Your proof is the constant itself.

Answer 6

show that a ball with one mass will fall from the same height and hit the ground in the same amount of time as a ball with greater mass
put two balls of different mass at the same distance above the ground and drop them both us v=v0+at to show that both at the same time after being dropped have the same v. use 0m/s as v0, 9.81m/s^2 as a, and 2s as t. This should give 19.62m/s as v.
Also you could just say that all the formulas for linear motion do not use mass meaning it is not needed.

Answer 7

Get two plastic baseballs. Poke a hole in one and fill with water. Drop both from the same height. They should land at the same time.