# Bodies falling in air

1. Jan 15, 2007

### 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

2. Jan 15, 2007

### Hootenanny

Staff Emeritus
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?

Last edited: Jan 15, 2007
3. Jan 15, 2007

### GlennB

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.

4. Jan 15, 2007

### Hootenanny

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

5. Jan 15, 2007

### GlennB

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

6. Jan 15, 2007

### Hootenanny

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

7. Jan 15, 2007

### GlennB

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)

8. Jan 15, 2007

### Hootenanny

Staff Emeritus
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.

9. Jan 15, 2007

### GlennB

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.

10. Jan 15, 2007

### DaveC426913

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.

Last edited: Jan 15, 2007
11. Jan 15, 2007

### DaveC426913

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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12. Jan 15, 2007

### Staff: Mentor

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.

13. Jan 15, 2007

### GlennB

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: )