Sears Tower Penny
Name: Gail S.
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. Help. Thank you so much. Gail S.
If you know the velocity of the penny and its mass, calculating it momentum
is easy; p = mv, where p is momentum, m is mass, and v is velocity. Force
is the change in momentum with time. So, to be able to determine the force
of impact, you need to know how long the impact lasts. This depends on a
number of factors, including the orientation of the penny and the hardness
of the surface it hits.
More relevant quantities to calculate would be the penny's momentum and its
kinetic energy. Momentum you already know how to calculate; kinetic energy
can be calculated in two ways: from the velocity, by E = mv^2/2, or from the
height fallen, by E = mgh, where g is the acceleration due to gravity ( =
9.8 m / s^2) and h is the height. This last formula can be used because the
kinetic energy of the penny at the bottom of its fall is equal to the
potential energy at the beginning. In fact, that's the method I would use to
calculate the velocity: first find the kinetic energy, and then calculate
the velocity from that. Or, more easily, you could factor out the redundant
mass terms ahead of time:
E = mv^2 / 2
E = mgh
mgh = mv^2 / 2
gh = v^2 / 2
v^2 = 2gh
v = sqrt(2gh)
Incidentally, when I use this formula, I calculate that the terminal
velocity is 93 m/s.
Richard E. Barrans Jr., Ph.D.
PG Research Foundation, Darien, Illinois
You need to know the acceleration that the penny will experience. The exact
value for this depends on what the penny hits. You may get a good estimate
by deciding how much time is required to stop the penny. A piece of
concrete will stop the penny in much less time than will dirt. In fact,
this is part of why landing on concrete hurts much more than landing in
dirt. The average acceleration is change of velocity divided by time
elapsed to stop the penny. I expect it will be less than a second for each.
This is a good opportunity to learn how to estimate.
Dr. Ken Mellendorf
Illinois Central College
Using energy conservation, I got the velocity to be 66
m/s, though I could be wrong. If you assume that the
penny hits a hard surface and deforms by 1mm, the
force would be 13,000 N. If you assume that the penny
hits on its edge, and that the average contact area
with the ground is 10mm square, then the
"Pressure*Travel" will be 1,300,000 N/m. If the
ground has a yield strength of, say, 20 KPSI (my
estimate for a human head), then the impact would
leave a dent 6cm deep.
Hi, Gail !!
Let us start calculating the velocity of the penny.
The potential energy at the top of the Tower is :
Ep = m.g.h = (0,003 kg)(10 m/s2)(443 m) = 13,29 Joules
The kinetic energy at the bottom is :
Ec = (1/2) mv^2 = (1/2)(0,003).v^2
Assuming that Ep = Ec then
13,29 joules = (1/2)(0,003).v^2
and finally : v = 94 m/s.
(how did you find 88 m/s ??)
When the body hits the soil some of the energy will
be used to change the physical structure of both the body
and the surface of the ground. Heat will be developed.
But, let us assume that nothing like mentioned happens.
Let us assume that a hole with a depth of 1 cm is
formed !! If it is so, than how big was the force to stop
the penny ?? We know that :
Vf^2 = Vi^2 - 2.a.X
0 = 94^2 - 2.a.(0,01) .: a = 441.800 m/s^2
F = m.a = (0,003)( 441.800 ) = 1325 N.
Well, the correct answer depends on so many factors, that only a more
detailed study can solve this question. I really do not know any formula
that could help us.
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Update: June 2012