Dropping a penny off a skyscraper.

[btpolk]

Homework Statement

A penny is dropped from a skyscraper.

a) Determine the terminal speed

b) How long will it take to reach terminal speed.

c) Will it exert enough force at terminal speed to break skin (460 psi is needed).

Homework Equations

a) F(air drag)=(1/2)(air density)v^2(surface area)(drag coefficient)

F= mg

c) conservation of energy?--P=F/A→Fd=(1/2)mv^2→P=(mv^2)/(2Ad)

The Attempt at a Solution

a) V(terminal)= √((2mg)/(air density*surface area*drag coefficient))=11.067 m/s

m= 0.0025kg air density=1.204 kg/(m^3) surface area= 0.000284m^2 drag coe=1.17

b) I completed an iterative process on excel and came up with t ≈11s d ≈122m

c) When I used the energy approach I get a pressure lower than that of when the penny is just laying on your skin. I'm not sure what went wrong. Oh and I'm assuming the penny is falling flat the whole way.

 

[cepheid]

c) conservation of energy?--P=F/A→Fd=(1/2)mv^2→P=(mv^2)/(2Ad)

How exactly did you figure out what d is here? I can't see how you'd have any way of knowing that.

 

[btpolk]

Product of terminal velocity and time it takes to reach terminal velocity.

 

[Dickfore]

how thick is the skin?

 

[btpolk]

460 psi is needed to break the skin.

 

[cepheid]

Product of terminal velocity and time it takes to reach terminal velocity.

This doesn't make too much sense to me. In this equation, "d" is the distance over which the work is done. Therefore, it is the distance over which the penny decelerates. In other words, it is the distance traveled between the moment of impact and the moment of coming to rest.

So what you really need to know is how far the skin is depressed. I imagine this depends on the elasticity of the skin. I really don't know how you are supposed to estimate it.

 

[btpolk]

Of course! Simple mistake. Thank you cepheid for your help!

My professor said I would have to make a number of assumptions in this problem, and I imagine the location of the point of contact on the body is one of them.

 

[Dickfore]

You can do it this way. Take the given P (remembering to convert the units), and calculate d? What do you get?

 

[btpolk]

If I do that I get 1.7e-4 m, but how does that help me find how much the penny is sinking in the skin?

 

[cepheid]

If I do that I get 1.7e-4 m, but how does that help me find how much the penny is sinking in the skin?

Well, I think what Dickfore might have been getting at is that you assume the same amount of depression for the case of the falling coin. This is kind of an upper limit on the amount of depression, and hence solving at least gives you a lower limit on the amount of force (and pressure).

 

[PhanthomJay]

If the falling penny hits you in the say the palm of your hand, you'd just feel a small sting, no puncture of the skin. I suppose if it hit you on your hard (bald!) head, it might cause a small skin puncture, although even that is questionable. I don't know how you would run the numbers without more data.

Now if a bullet was dropped, all bets are off, because of its high terminal velocity and small contact area.

 

[cepheid]

You might also be interested in listening to the very last story at the bottom of the page I'm linking to below

http://www.cbc.ca/quirks/episode/2012/03/24/march-24-2012/

 

[btpolk]

I'm not sure I understand. So can I use this to prove that the penny can't exert that much force?

 

[btpolk]

If the falling penny hits you in the say the palm of your hand, you'd just feel a small sting, no puncture of the skin. I suppose if it hit you on your hard (bald!) head, it might cause a small skin puncture, although even that is questionable. I don't know how you would run the numbers without more data.

Now if a bullet was dropped, all bets are off, because of its high terminal velocity and small contact area.

I do realize that it won't do a lot of damage, if any, but I do have to prove it mathematically.

 

[PhanthomJay]

You might also be interested in listening to the very last story at the bottom of the page I'm linking to below

http://www.cbc.ca/quirks/episode/2012/03/24/march-24-2012/

Thanks, that's good news. Some 50 years ago when i was a boy, barely tall, I chucked a penny, when my parents weren't looking, through the railings of that majestic building's 86th floor outdoor observation deck. I didn't know a hoot about Physics, but somehow I surmised that it would do no harm. The years came later when I became a man, and i wondered...oh how I wondered ...if I had hurt anyone that day. Now, I can rest.

btpolk: You say you can make assumptions, so assume the penny falls flat on a person's noggin', and it decelerates to a stop in a mere .001 second. That will allow you to calculate the average impact force, using the impulse-momentum relationship, from which you can calculate the pressure by dividing the impact force by the surface area of the penny, and compare it to the 450psi allowable. You should find that the pressure is quite small, even if he impact time is smaller than the asumed value.