

Sears Tower Penny
Name: Gail S.
Status: educator
Age: 50s
Location: N/A
Country: N/A
Date: 20012002
Question:
My 7th grade classes are currently studying Newton's Laws
of Motion and we got into a discussion about falling objects. How do we
go about calculating the force that a penny would have when it hit the
pavement if dropped from the top of the Sears Tower? The tower is about
443 meters high and the penny has a mass of about 3 grams. We figured
that by the time it hits the ground (pretending no air resistance) it is
falling at a nearly 88 m/sec, but from there we are not sure how to
calculate (what formula to use) for the actual force that it will have
when it hits.
Replies:
To do this calculation, you need a very clear understanding of what
"force" is. Force is the rate at which momentum changes, so if the
penny is decelerated rapidly, the force it exerts will be larger (and
will be exerted for a shorter time) than if the penny were decelerated
gradually. In other words, the penny does not "have" a force; what it
has is momentum, and it can "spend" this momentum by exerting force,
but the force it exerts depends partly on the properties and motion of
whatever it collides with.
But that is not very satisfying. We still want some sense of how hard
this thing will hit. I mean, if we were down on the street below,
should we run for our lives from the path of this penny, or try to
catch it?
We can get a sense by answering two simpler questions: 1) how much
momentum does the penny have? and 2) at what rate is the penny likely
to be decelerated?
So, how much momentum does the penny have, in terms a person might have
some feeling for? Not much, actually. It has .264 kgm/s of momentum,
about the same as a can of soup rolling at babycrawling speed (.5
m/s). Not a scary amount of momentum, then, but should you try to catch
the penny? Let us see how rapidly this momentum might be spent.
The penny is moving so much faster than your hand is likely ever to
move that it is a reasonable first approximation to say that your hand
will remain stationary as you try to catch the penny. Let's say the
penny has to stop in 1 cm, and see what force would be exerted if it
were to decelerate at a constant rate from 88 m/s to 0 in this
distance.
The average speed during deceleration is 44 m/s, and it travels .01 m,
so it decelerates for 230 microseconds. So, the force during this time
is the total momentum change divided by the time during which the
momentum is changing:
.264/2.3e4 = 1161 kgm/s/s.
This force is what it would take to hold up a mass of 118 kg in Earth's
gravity  about 250 lbs.
That is going to hurt. You could conceivably catch it, but it is going
to break some bones in your hand if you are lucky and it lands flat. If
it lands edge on, it is going through. If it were to hit you in the
head, it clearly cannot take 1 cm. to decelerate  more like 1 mm. In
this case, the average force would be over a ton.
Tim Mooney
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Update: June 2012

