# Bodies falling in air

• GlennB

#### GlennB

Hi all - I wonder if you could settle a dispute that's happening at a non-science forum? The maths itself seems complex, but the principle will be very familiar.

Two bodies are identical in every respect, except for mass (e.g. a solid steel ball and a hollow steel ball of the same dimensions)

They are dropped simultaneously, through the air, from a height such that neither reaches terminal velocity.

Does the heavier ball "pull away" from the lighter one in this scenario i.e. fall faster?
Or would it only "pull away" after the lighter one had reached terminal velocity?

GlennB

Welcome to the forums,

If neither reaches the terminal velocity, neither will #significantly# pull away from the other. However, the terminal velocity is proportional to the square root of the mass of the object; therefore, the lighter mass would reach its terminal velocity before the heavier one and hence the heavier one would "pull away" from the lighter one. This assumes of course that they are dropped from a sufficient height to allow the lighter object to reach its terminal velocity.

Does that make sense?

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Welcome to the forums,

If neither reaches the terminal velocity, neither will pull away from the other. However, the terminal velocity is proportional to the square root of the mass of the object; therefore, the lighter mass would reach its terminal velocity before the heavier one and hence the heavier one would "pull away" from the lighter one. This assumes of course that they are dropped from a sufficient height to allow the lighter object to reach its terminal velocity.

Does that make sense?

We'd dug the formula for terminal velocity out of Wikipedia, so understood that it was proportional to mass.
What seems counter-intuitive is that mass is not a factor when measuring the momentary speed of a "still accelerating" body in a fluid.

What seems counter-intuitive is that mass is not a factor when measuring the momentary speed of a "still accelerating" body in a fluid.
Ahh, but what makes you say that mass isn't a factor?

Ahh, but what makes you say that mass isn't a factor?

Sorry.
I should perhaps have said 'when comparing the momentary speeds of 2 "still accelerating" differently massive but otherwise identical bodies' ? ;)

Sorry.
I should perhaps have said 'when comparing the momentary speeds of 2 "still accelerating" differently massive but otherwise identical bodies' ? ;)
Still, why do you think that mass isn't a factor?

Still, why do you think that mass isn't a factor?

That's based on your original reply, indicating that the two bodies fall together until the lighter one reaches terminal velocity. Maybe I misunderstood your post.

(until today I would have thought the more massive one would 'outpace' the lighter one - in a fluid - even before terminal velocity is reached by either)

Apologies, in my original reply, I should have said neither would "significantly" pull away since the height over which they are dropped would be relatively small. However, the mass of an object does have affect the acceleration and hence the instantaneous velocity of a projectile. My bad.

Apologies, in my original reply, I should have said neither would "significantly" pull away since the height over which they are dropped would be relatively small. However, the mass of an object does have affect the acceleration and hence the instantaneous velocity of a projectile. My bad.

No problem. And, in fact, it got me to thinking that my "intuition" was maybe right after all (though I totally accept that intuition and physics don't always get along too well). It goes like this :

There is a final "instant" where the lighter body stops acclerating and reaches terminal velocity. The heavier object continues accelerating beyond this point.
If their momentary speeds are matched up until this point, the heavier object would then need to start accelerating *at a greater rate* from this point on. Which is ridiculous.

... neither would "significantly" pull away since the height over which they are dropped would be relatively small. However, the mass of an object does have affect the acceleration and hence the instantaneous velocity of a projectile...

Really? You're saying that a light ball and a heavy ball would almost pace each other all the way up until the light ball hit terminal velocity?

Oh, I see, over short distances - < a few dozen feet, i.e. nowhere near terminal velocity.

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There is a final "instant" where the lighter body stops acclerating and reaches terminal velocity. The heavier object continues accelerating beyond this point.
If their momentary speeds are matched up until this point, the heavier object would then need to start accelerating *at a greater rate* from this point on. Which is ridiculous.
Why do you say that? It would continue accelerating at the same rate from that point on. The lighter one's acc. will have dropped to zero.

(1st diag. ideal case; 2nd diag. more realistic but less clear)

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The acceleration curve for both objects are hyperbolic, but different in their limit, so except for the instant that they are dropped (and after both are at terminal velocity), the denser object is always accelerating faster than the less dense object.

Why do you say that? It would continue accelerating at the same rate from that point on. The lighter one's acc. will have dropped to zero.

(1st diag. ideal case; 2nd diag. more realistic but less clear)

What I meant was that if the 2 balls fell at the same rate until the light one hit terminal velocity, the heavier one would now be in a bit of trouble in achieving its term.vel. The graph would have a kink in it, which is ridiculous, therefore the original (incorrect) proposition is daft.

(if you see what I mean :yuck: )