# Two balls dropped from the same height

I have seen youtube videos where two balls of the same size and of different masses are dropped from the same height and they hit the ground at the same time.

I understand why this is so: the increase in the mass of a body increases the force of gravity acting on the body, but also decreases the body's willingess to move.
It makes sense to me in vacumm, but it somehow collides with what i have found written in my textbook. It goes more less like this:

If you dropped an ant from a certain height in the air, it would not die upon landing, because the increase in speed as it is falling entails the increase of air resistance acting on the ant. The force of air resistance quickly balances the force of gravity acting on the ant. Up to this point the ant hasn't gained much speed and now is no longer accelerating. The speed is too little to cause it much harm when landing.

Then it says that if you dropped a human from that height it would also stop accelerating at some point (at the point of air resistance balancing the body's weight), but it would happen much later. Therefore a human would gain more speed before landing and would die.

This brings me back to the experiment with two balls.
Why doesn't the same principle work here?
I know their shape makes air resistance less powerful but it must still be there.
The lighter ball has lesser weight, so it should take less time for air resistance to counter the ball's weight than it should take in the case of the ball with greater mass.
The heavier ball should accelerate longer then the lighter ball, and therefore it should hit the ground first.

Please tell me, where is the error in my thinking.

## Answers and Replies

A.T.
Science Advisor
It makes sense to me in vacumm
It's only true in vacuum.

hipokrytus
anorlunda
Staff Emeritus
You are correct, there. Is some air resistance, even for balls. It might be clearer for you if you imagine the metal in the smaller ball being hammered into the shape of a feather. With the bigger surface area, an iron feather will experience even more air resistance.

hipokrytus
BvU
Science Advisor
Homework Helper
No error, you are absolutely right: the experiment is to be conducted in vacuum to eliminate the influence of air resistance.

I'm not sure your argument
The lighter ball has lesser weight, so it should take less time for air resistance to counter the ball's weight
is the correct way to say it (it's not the weight, but the air resistance that counts), but the terminal velocity is more or less proportional to the square root of the radius.

hipokrytus
Thank you for your responses.
The experiment was conducted in the open air, and the performer lifted the balls as high as his hands could go up (so not very high).
So is it correct to conclude that at this height both balls were accelerating all the way?
And if the height were (much) greater we would notice difference in time in which they hit the ground?

sophiecentaur
Science Advisor
Gold Member
The lighter ball has lesser weight, so it should take less time for air resistance to counter the ball's weight

It isn't really a matter of time. The fact is that the air resistance does not depend on the actual masses of two objects of the same size and shape. But the weight force is, of course, greater for the more massive object. (Just putting it a different way, anorlunda)

hipokrytus
I think i phrased that badly.
What I said there refers to the fact that air resistance increases if the velocity of a falling body icreases too.
And when this force of air resistance reaches a point where it balances the weight force of the body, the acceleration ceases, and the body is from then on falling with constant speed.
The time during which the ball with lesser weight is accelerating is shorter when compared to the time during which the heavier ball is accelerating because there is less weight force for the air resistance to balance (in the case of the lighter ball). Is that correct?
I may have phrased it even worse now.
- , -

PeroK
Science Advisor
Homework Helper
Gold Member
2021 Award
I think i phrased that badly.
What I said there refers to the fact that air resistance increases if the velocity of a falling body icreases too.
And when this force of air resistance reaches a point where it balances the weight force of the body, the acceleration ceases, and the body is from then on falling with constant speed.
The time during which the ball with lesser weight is accelerating is shorter when compared to the time during which the heavier ball is accelerating because there is less weight force for the air resistance to balance (in the case of the lighter ball). Is that correct?
I may have phrased it even worse now.
- , -

Imagine two people of the same mass, shape and size jump out of an aircraft, but one has a parachute, which is then opened. Who falls faster? The one with the parachute is undoubtedly heavier.

Also, if there is a breeze blowing, leaves and bits of paper may be blown upward, but it would take a ferocious wind to blow a human being up into the air. Or consider an aircraft, moving at its normal flying speed. Why doesn't it fall?

You're confusing two things here:

1) the simple law of gravity, whereby all objects are given the same acceleration by gravity.

2) The complex behaviour of objects of various sizes and shapes within the atmosphere, which includes thermals and air currents as well as air resistance and the aerodynamic principles by which birds, aircraft and helicopters can fly.

hipokrytus
anorlunda
Staff Emeritus
I think i phrased that badly.
What I said there refers to the fact that air resistance increases if the velocity of a falling body icreases too.
And when this force of air resistance reaches a point where it balances the weight force of the body, the acceleration ceases, and the body is from then on falling with constant speed.
The time during which the ball with lesser weight is accelerating is shorter when compared to the time during which the heavier ball is accelerating because there is less weight force for the air resistance to balance (in the case of the lighter ball). Is that correct?
I may have phrased it even worse now.
- , -

Each body will accelerate until reaching terminal velocity.

The higher the ratio of surface area to mass, the lower the terminal velocity.

The acceleration will decrease asymtotically until reaching zero at terminal velocity. In theory it could take infinite time to reach exactly terminal velocity.

Therefore, the time to reach terminal velocity is harder to define. It couild be infinite.

PeroK
Science Advisor
Homework Helper
Gold Member
2021 Award
Therefore, the time to reach terminal velocity is harder to define. It couild be infinite.

Except:

a) Gravity varies with height.

b) Air pressure varies with height.

c) The air is, in any case, not homogeneous and inert.

d) You'll hit the ground sooner or later.

anorlunda
Staff Emeritus
Except:

a) Gravity varies with height.

b) Air pressure varies with height.

c) The air is, in any case, not homogeneous and inert.

d) You'll hit the ground sooner or later.

Even with all those excepts, the time to reach terminal velocity is harder to define. But I agree, if you hit the ground it won't be infinite.

EDIT: I take all that back. PeroK is pointing out that terminal velocity could decrease with altitude because of air pressure, then approach will not by asymtotic.

hipokrytus
A.T.
Science Advisor
So is it correct to conclude that at this height both balls were accelerating all the way?
Yes, but they accelerate, and thus fall slightly differently even here.