Air Resistance Question -- Dropping a tennis ball and a feather....

[theanswer2physicsisu]

Homework Statement

Which of the following is true regarding a tennis ball and a feather dropped from the same height? Assume that air resistance acts on both objects:
A. The tennis ball has more mass so it will fall faster.
B. The tennis ball has more surface area so it will fall faster.
C. The tennis ball has less surface area so it will fall faster.
D. The tennis ball has more weight so it will fall faster.

Relevant Equations

No relevant equations that I know of except maybe air resistance

I thought the one with more mass would fall faster

 

[haruspex]

No relevant equations that I know of except maybe air resistance

So what equation can you write involving air resistance and acceleration?

Btw, none of the answers is correct.

 

[PeroK]

E. None of the the above.

 

[hmmm27]

Look at the options. Which one(s) (or, as the above posters are trying to make you believe, not a single one) make sense to you.

 

[phinds]

I thought the one with more mass would fall faster

Why would that be true in general? Do you think a 2oz feather would fall faster than a 1oz ball bearing?

 

[berkeman]

Btw, none of the answers is correct.

E. None of the the above.

Hmm, I got the right answer. Remember, air resistance is involved...

 

[phinds]

Hmm, I got the right answer. Remember, air resistance is involved...

You lost me on this one. How do you figure one of those answers is correct? Taken as general statements, it is easy to give counter examples to each of them.

 

[berkeman]

You lost me on this one. How do you figure one of those answers is correct? Taken as general statements, it is easy to give counter examples to each of them.

Check PMs...

Spoiler

Think chubby bicyclist on a downhill faster than you can pedal compared to a very light bicyclist with approximately the same surface area/air resistance...

 

[pbuk]

If you think the answer is C do you think that a feather with a surface area of 1 square centimetre (which is less than that of a tennis ball) will fall faster?

 

[berkeman]

If you think the answer is C

Who is "you"? The Reply/Quote feature comes in handy sometimes...

 

[pbuk]

Who is "you"?

Anyone that thinks the answer is "C".

 

[berkeman]

Okay, okay. We're clearly not going to help the OP if we keep kidding around. If anybody disagrees with my spoiler hint, please let me know by PM. Otherwise, the OP's answer in post #1 looks correct to me.

 

[berkeman]

And for way more background information, we have this:

https://www.physicsforums.com/threads/falling-objects-tennis-ball-vs-feather.825156/

For this question in this thread, it's probably safe to assume that the $C_d$ drag coefficients of the feather and tennis ball are similar, and the mass of the tennis ball is several orders of magnitude greater.

 

[berkeman]

Apologies, it looks like I managed to lock this thread yesterday -- I don't think I meant to do that. Thread is unlocked again.

You lost me on this one. How do you figure one of those answers is correct? Taken as general statements, it is easy to give counter examples to each of them.

In EMS (EMT, Paramedic) exams, often you have to pick the best alternative of the choices given, even though each one may have issues on its own. That's what I did in this case, since many bird feathers have a $C_d$ that is close to a fuzzy tennis ball.

 

[PeroK]

The correct answer is that the tennis ball falls faster because the total external force on the ball per unit of mass is greater than that of the feather. In other words, Newton's second law. The tennis ball falls faster because:
$$\frac{m_tg - f_t}{m_t} > \frac{m_fg - f_f}{m_f}$$where $f_t, f_f$ is the air resistence on the ball and feather respectively. If we take the weight of an object to be $mg - f$, then there is nothing to say that a large feather weighs less than a small tennis ball. Weight (however you define it) does not determine acceleration.

 

[bob012345]

If we take the weight of an object to be $mg - f$, then there is nothing to say that a large feather weighs less than a small tennis ball. Weight (however you define it) does not determine acceleration.

Yes, a parachute could be thought of as similar to a very large feather in this context.

 

[DrJohn]

Isn't it the one with the higher density.
Think of a table tennis ball, a wooden ball and a solid steel ball, all exactly the same diameter, on a sloping ramp. Which will have the higher terminal velocity (Assuming slope is long enough)?

I use this example to explain why, if you have two identical gliders/sailplanes, but one has its wings full of, say 100kg of water in each one, then the heavier glider will fly down the same slope faster. (For any other pilots, the heavier one will have its best glide at a higher speed).

Note that there is a little bit more required to explain this, but that's what your problem is about.

 

[pbuk]

Isn't it the one with the higher density.

That is not an answer to the question in the OP, but if you are asking "will an object with a higher density fall faster than an object with a lower density" then consider a bottle full of water suspended under a parachute and a bottle half full of water with no parachute. Which one has the higher density? Which one falls faster?

 

[pbuk]

The tennis ball falls faster because:
$$\frac{m_tg - f_t}{m_t} > \frac{m_fg - f_f}{m_f}$$where $f_t, f_f$ is the air resistence on the ball and feather respectively.

Note that this simplifies to $g - \dfrac{f_t}{m_t} > g - \dfrac{f_f}{m_f}$, or even simpler $\dfrac{f_t}{m_t} < \dfrac{f_f}{m_f}$.

So the correct answer is "the tennis ball falls faster because it has less air resistance in relation to its mass than the feather", or in the terminology of the OP and noting that air resistance is closely related to surface area, "the tennis ball falls faster because it has less surface area in relation to its mass than the feather".

 

[DrJohn]

That is not an answer to the question in the OP, but if you are asking "will an object with a higher density fall faster than an object with a lower density" then consider a bottle full of water suspended under a parachute and a bottle half full of water with no parachute. Which one has the higher density? Which one falls faster?

Changing the question half way through to make my answer look wrong is really just a bad explanation.
Higher density was a hint to help work out the answer - OP has to make an intelligent guess to which one has the higher density.

For a start, you are deliberately ignoring the mass of the parachute in your rebute (the ones I had for my aircraft were about 15lbs, the one I had on my back on Saturday and Sunday was 18lbs, but I didn't have to jump out, of course)
You are ignoring the volume occupied by the parachute.
You are well aware that the shape of the parachute influences the drag, and I bet you know that a big parachute that doesn't open properly and gets entangled with itself will probably mean the user falls faster than someone using a parachute with 10% smaller total surface area but fully opened. (unless we are talking about tiny parachutes that just will not work)
You are ignoring that as a pilot I am very well aware of the effect of drag in its various types. Your answer is aimed directly at form drag (one type of parasitic drag) - caused by the shape of the object. Form drag, as we all know, increases with speed

Then you reach the same answer as me and say

"the tennis ball falls faster because it has less air resistance in relation to its mass than the feather", or in the terminology of the OP and noting that air resistance is closely related to surface area, "the tennis ball falls faster because it has less surface area in relation to its mass than the feather".

Which is saying the object with the greater density has less drag per unit volume, although the shape and hence drag will complicate things a little bit.

So have a read of my example featuring table tenis ball, wooden ball and steel ball. I know that you will get the correct answer to which will reach the highest velocity, assuming the slope is long enough for all the objects to reach their teminal velocity. But even on a medium length slope, say 10 or 15 or so metres in length, with no ball carrying a parachute just to make things clearer, we both know which one will be traveling fastest at the 10 metre mark - "The one with less air resitance in relationship to its mass". The most dense of the three.Now here's an interesting one, where many glider pilots get the right answer to the first half but give the wrong reason for their answer, but I suspect you will get it correct.

Two identical gliders (ie same model, same shape, same empty mass), with identical weight pilots are flying at the same speed (we will say fast, say 100 knots, to make this more straightforward). And are flying together at the same height. But one glider has 100kg of water ballast in each wing, the other has no water ballast on board (not any other type of ballast). Both pilots pull back on the control column by the same amount / same angle / same stick movement at the same 100 knots speed, and not at an angle that risks stalling the aircraft due to excessive g loading and huge angle of attack. And they dont' adjust the glider's flaps, either. I think I've covered all excuses or silly things and variations that would mess things up.
So.
Which one will pull up by the largest amount, and why?
(Or do they both pull up to the same height?)

Remember PE = mgh and KE = 1/2mv²
h = height of course.

Gone on, give it a try, but the reason why is the important bit ;)

 

[jbriggs444]

Higher density was a hint to help work out the answer - OP has to make an intelligent guess to which one has the higher density.

Cross-sectional area is more relevant than volume. The sort of "density" that you want is mass per unit of cross-sectional area in the direction of fall.

A steel wrecking ball has a higher terminal velocity than a spherical fleck of steel dust, despite having the same mass per unit volume. This is a consquence of the cube-square law.

 

[hutchphd]

Then you reach the same answer as me and say

I would just say "shape matters"...

 

[haruspex]

That is not an answer to the question in the OP

Changing the question half way through

I believe @pbuk was merely pointing out that greater density is not one of the offered answers. This was not changing the question.

noting that air resistance is closely related to surface area

Then you reach the same answer as me

Not the same. The dimensional difference between volume and area is crucial.