Why does the last term detract from acceleration?

[Peter G.]

I'd like to start off by saying that all I want is explanations. My head forces me to think in a way that goes against what I "know" is right:

Firstly: I "know" all objects fall at the same speed in a vacuum. What I understand from a vacuum is the absence of air, thus, absence of air friction - air resistance, in other words, an absence of the force pulling things upwards. The force pulling us down, as shown in several parachute cartoons is weight: W = m x g. If there is no force pulling us upwards and the only force acting on us is our weight, dependent on mass, why don't heavier objects fall faster in a vacuum?

The same concept confuses me when air resistance is involved. I remember taking my IGCSE's and facing a question regarding two balls, one made of aluminum and another one made out of plastic. Both had the exact same shape, but naturally, different weights. The answer said that both balls would fall at the same rate. But, once again I am confused. Having the same shape, they have the same aerodynamic properties, meaning the air resistance acting on them will be the same at corresponding speeds, but, one ball is heavier than the other, meaning the force pulling it down is greater. So, shouldn't it fall faster?

I hope what I wrote is clear.

Thanks,
PeterG

 

[capandbells]

I'd like to start off by saying that all I want is explanations. My head forces me to think in a way that goes against what I "know" is right:

The force of gravity is greater for a more massive object, but acceleration due to gravity is independent of mass.
The force due to gravity is F = mg. By Newton's second law, F = ma or a = F/m = mg/m = g.

 

[Doc Al]

I'd like to start off by saying that all I want is explanations. My head forces me to think in a way that goes against what I "know" is right:

Firstly: I "know" all objects fall at the same speed in a vacuum. What I understand from a vacuum is the absence of air, thus, absence of air friction - air resistance, in other words, an absence of the force pulling things upwards. The force pulling us down, as shown in several parachute cartoons is weight: W = m x g. If there is no force pulling us upwards and the only force acting on us is our weight, dependent on mass, why don't heavier objects fall faster in a vacuum?

In a vacuum, all objects have the same acceleration. Given the force, how do you find the acceleration? Use Newton's 2nd law, F = ma. In this case, the force is the weight = m x g. So set that force equal to m x a. Thus: m x g = m x a. The masses cancel and you get a = g, independent of mass.

In words: Yes, the force of gravity is proportional to mass. But the acceleration for a given force is inversely proportional to mass. It cancels nicely.

The same concept confuses me when air resistance is involved. I remember taking my IGCSE's and facing a question regarding two balls, one made of aluminum and another one made out of plastic. Both had the exact same shape, but naturally, different weights. The answer said that both balls would fall at the same rate. But, once again I am confused. Having the same shape, they have the same aerodynamic properties, meaning the air resistance acting on them will be the same at corresponding speeds, but, one ball is heavier than the other, meaning the force pulling it down is greater. So, shouldn't it fall faster?

Sure, the heavier object will have the greater acceleration if you include air resistance. (No idea why the 'answer' said different, unless they were ignoring air resistance.)

Again, use F = ma to see this. For a given velocity, the force of air resistance will be the same; let's call it Fair. The net force on the ball will be F = mg - Fair. The acceleration will thus be F/m = g - Fair/m. The bigger the mass m, the greater the downward acceleration. (The second term will be smaller.)

Make sense?

 

[Peter G.]

Thanks a lot both of you!

Regarding the acceleration in a vacuum, it is very clear.

This is the only thing that confuses me: F/m = mg/m = g, therefore: g - Fair/m. I don't understand where the m comes from

Edit: Sorry, read your post. Got it now! Thanks for the patience!

I am tired of my books saying this and that and not explaining it! It basically, implicitly asks us to memorize stuff when I like to learn!

 

[Doc Al]

With air resistance though, the only thing that is confusing me is this: " F/m = g - Fair/m"

Can you pinpoint what's confusing you about it?

Here's how it's derived:
Forces on ball: mg down & Fair up
Net force on ball: ΣF = mg - Fair (taking down as positive)
Applying ΣF = ma: mg - Fair = ma
Solving for a: a = (mg - Fair)/m = g - Fair/m

That last term, Fair/m, detracts from the acceleration due to gravity. The smaller it is, the closer the acceleration is to g. And the bigger the mass m is, the smaller is Fair/m.