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I'm far from being a physics expert and figured this would be a good place to ask a beginner question that has been confusing me for some time.

According to Galileo, two bodies of different masses, dropped from the same height, will touch the floor at the same time in the absence of air resistance.

BUT Newton's second law states that $a = F/m$, with $a$ the acceleration of a particle, $m$ its mass and $F$ the sum of forces applied to it.

I understand that acceleration represents a variation of velocity and velocity represents a variation of position. I don't comprehend why the mass, which is seemingly affecting the acceleration, does not affect the "time of impact".

Can someone explain this to me? I feel pretty dumb right now :)

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    $\begingroup$ Minor caveat for VERY heavy masses: physics.stackexchange.com/q/3534/2451 $\endgroup$
    – Qmechanic ♦
    Commented Jul 5, 2011 at 19:52
  • $\begingroup$ You are right to think of neglecting air resistance, but you also have to neglect air buoyancy due to Archimedes' principle. This is also an is easily observed effect by setting the right conditions. $\endgroup$
    – babou
    Commented Jun 12, 2013 at 6:06

7 Answers 7

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Newton's gravitational force is proportional to the mass of a body, $F=\frac{GM}{R^2}\times m$, where in the case you're thinking about $M$ is the mass of the earth, $R$ is the radius of the earth, and $G$ is Newton's gravitational constant.

Consequently, the acceleration is $a=\frac{F}{m}=\frac{GM}{R^2}$, which is independent of the mass of the object. Hence any two objects that are subject only to the force of gravity will fall with the same acceleration and hence they will hit the ground at the same time.

What I think you were missing is that the force $F$ on the two bodies is not the same, but the accelerations are the same.

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it is because the Force at work here (gravity) is also dependent on the mass

gravity acts on a body with mass m with

$$F = mg$$

you will plug this in to $$F=ma$$ and you get

$$ma = mg$$ $$a = g$$

and this is true for all bodies no matter what the mass is. Since they are accelerated the same and start with the same initial conditions (at rest and dropped from a height h) they will hit the floor at the same time.

This is a peculiar aspect of gravity and underlying this is the equality of inertial mass and gravitational mass (here only the ratio must be the same for this to be true but Einstein later showed that they're really the same, i.e. the ratio is 1)

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    $\begingroup$ This is not a good answer does not explain things from first principles unlike the answers which start with the $F = \frac{GM}{R^2} \times m$ equation. $\endgroup$
    – Autodidact
    Commented May 25, 2019 at 22:32
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    $\begingroup$ The first eqn is what we are trying to prove.That for every body it’s a=g.That you assumed is true from start. And then everyone can do the math that g is in place of a. $\endgroup$
    – cOnnectOrTR 12
    Commented Jul 26, 2021 at 19:22
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There are two ways that mass could effect the time of impact:

(1) An object which is very massive has a stronger attraction to the earth. Logically, this might make the object fall faster and so reach the ground sooner.

(2) An object which is very massive is difficult to get moving. (I.e. it has very high inertia.) Thus one might logically expect the very massive object to be more difficult to get moving and so to lose the race.

The miracle is that in the world we live in, these two effects exactly balance and so the heavier mass reaches the ground at the same time.

Now let me give a simple explanation for why it's natural that this comes about. Suppose we have two very heavy masses. If we drop them separately they take some time T to fall. On the other hand, if we attach them together, will they take the same length of time? Think about a sphere split into two halves:

Which falls faster: a sphere or a sphere split into two halves?

The two halves of the sphere would fall at the same speed as each other. So if you dropped them next to each other, they'd fall together. And dropping them next to each other isn't going to be any different from screwing them together and dropping them together. That is, there won't be any force on the screws. So the combined (or screwed together) sphere has to fall at the same rate as the split sphere.

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    $\begingroup$ The beautiful explanation in the second half of your answer is more-or-less the same thought experiment which led Galileo to make his bold claim that dropping a heavy and a light ball from the leaning tower of Pisa they would reach the earth at the same time - contrary to what Aristotle had claimed. $\endgroup$
    – Nadav Har'El
    Commented Jun 29, 2021 at 21:39
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Because force 'pushing' object closer to earth is proportionally bigger for 'heavier' object. But heavier object is also have higher gravitation force.

So these two factors perfectly compensate each other: Yes you need more force for a set acceleration, but more force is here due to heavier mass.

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Lets say two separate mass $M_1$ and $m_2$ where $M_1$ >> $m_2$, both fall, from the same instant in a gravitational field

Force on $M_1$ is $F1$ = G $M_{\text{earth}}$ $M_1$/ $R^2$

Force on $m_2$ is $F2$ = G $M_{\text{earth}}$ $m_2$/ $R^2$

Therefore the forces are $F1$ >> $F2$

So, most people think $M_1$ should accelerate much faster than $m_2$

But as you wrote above a = F/m and substituting $F1$, $F2$, $M_1$ and $m_2$ into that formula we find:

$F1$/$M_1$ = $F2$/$M_2$ = G $M_{\text{earth}}$/ $R^2$

Therefore the acceleration is independent of the masses we drop, and is a constant.

EDIT Bother, by the time I had written this down, it was answered, obviously the other authors had a different acceleration in their typing.

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  • $\begingroup$ What happens when M1 >> m2 is false? Does m2 pull on M1 mean acceleration grow with m2? $\endgroup$
    – Lorenzo Boccaccia
    Commented Mar 7, 2014 at 1:04
  • $\begingroup$ I wonder if there is an "intuitive" GR way of explaining this. One could say that in the model of a large mass warping space-time that only the distance from the center of mass that creates the disturbance effects the acceleration of another mass toward it's center, but that doesn't really say "Why?". Also, if this wasn't the case, we couldn't predict orbital periods without knowing the mass of the orbiting object. Is the best we can do is say that the answer to "Why?" is "Because that is what we observe."? $\endgroup$
    – Jack R. Woods
    Commented Apr 2, 2018 at 16:35
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Let us think about this by contradiction.

Suppose the two masses fall at different rates (say, heavier mass falls faster), then if you tie the two masses together, what will happen?

Solution #1. if you tie the masses together, they form a even larger mass, thus they fall faster

Solution #2. if you tie the masses together, the lighter mass will give the heavier mass a drag force, thus they fall slower.

The two solutions contradict each other; so they must fall at the same rate.

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  • $\begingroup$ Nothing requires #2 to be true. Drag force only depends on area, shape velocity and fluid medium. It's possible to put two objects together in a way that doesn't increase the drag force at all (and due to the increased weight, increases the terminal velocity). You can also take an object of the same mass and either drop it at a new angle, or change it's shape, and it will fall at a different speed with the same mass. The contradiction you point out isn't really a contradiction. $\endgroup$
    – JMac
    Commented Jun 25, 2019 at 18:52
  • $\begingroup$ well I agree its not rigorous... thank you for pointing out :) $\endgroup$
    – Zecheng Gan
    Commented Jul 2, 2019 at 21:08
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I am answering this in my own way critics are welcomed.

So, Generally consider two bodies of same shape,surface area (same) but MASSES DIFFERENT.

Now if you release both the bodies from a similar height say h, with u =0m/s for both (initially rest or aka free fall) they travel h height in same time .

So,what is happening here is we know Newton's law of gravitation like F = (GMm)/R^2

Where, G is Grav.const. and M and m are masses along which gravitation (attraction only always) acts and R is the distance between the center of masses or simply centers of body (We just take point objects usually dw bout it).

Now F=[(GMē) m]/R^2 Mē= earth mass and R dist.between earth center and object

You can simplify (GMē/R^2) as g' usually when distance of object from earth surface is very less than earth radius you can take g'=g =9.8m/s^2

So,finally F=gm=mg

For two different masses This F so called gravitational force of attraction is different but look g is same for both masses that is acceleration of two bodies is same even though there is different mag.of force on them .

Mostly ppl are confused here they generally interpret this as greater the force greater the acc.n which is not right to be honest

Greater force doesn't mean greater acc.n greater force maybe because of their masses being different but with same acc.n so it is possible for two bodies of different masses experiencing different forces having same acc.n

It is more right if you put this explanation of so called g being same for two bodies rather than a1=a2=g YOU BETTER USE a1=F1/m1 and a2=F2/m2 and then you can see that g=F1/m1=F2/m2 You will be getting a lil clarity if you do it this way .

Therefore since same acc.n it says both bodies have same change in their velocities over equal time interval so these bodies attain same final velocities over a particular time since they started with 0m/s at same time that's why they cover equal disp.during their journey down hence they reach ground in same time.

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    $\begingroup$ please use Mathjax for any math related content for formatting $\endgroup$
    – LPZ
    Commented Jul 11, 2022 at 15:19
  • $\begingroup$ Please don't use weird abbreviations. $\endgroup$
    – PM 2Ring
    Commented Nov 14, 2023 at 1:19

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