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Name: Gail S.
Status: educator
Age: 50s
Location: N/A
Country: N/A
Date: 2001-2002


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 kg-m/s of momentum, about the same as a can of soup rolling at baby-crawling 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:

.264/2.3e-4 = 1161 kg-m/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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