What would happen if you dropped a superball off the Empire State Building?

As soon as the ball is dropped, it starts to accelerate of course, due to the force of gravity. However, the faster it goes, the more important the effects of air resistance become. After the ball has dropped about 100 meters (around 25-30 stories for a normal building), the force of air resistance becomes large enough to balance the force of gravity; at this point, the ball has reached its "terminal velocity", and falls at a constant rate without further acceleration. Plugging in the numbers for a typical superball (one of the big ones that are about two inches in diameter, not the dinky ones you get from a vending machine), we find that the terminal velocity is 70 mph. (The terminal velocity for a small superball would be only about 50 mph.) That's how fast the ball will be going when it hits the sidewalk at the bottom of the Empire State Building. This would be considered a rather slow pitch in baseball, so that if it hit you on the shoulder, it would sting pretty badly, but wouldn't cause any serious injury. Clearly, the ball will not shatter at this speed. (A reasonable high-school baseball pitcher can throw this fast.)

Assuming it hits the pavement, the superball will rebound with about 85% to 90% of its impact velocity; this is determined by the "coefficient of restitution", which is of course quite high for a superball. This means that the rebound velocity will be about 60 mph. Including effects of air resistance, the rebound height will be about 80 feet, or about 7 stories. Because the terminal velocity on the way down is reached after about 25 stories worth of drop, the rebound height would be the same if you dropped the superball from a 25-story building, instead of the Empire State Building (which is 102 stories high).

Note that the rebound height is much less than 25 stories, because most of the acceleration on the way down occurs in the first 10 stories or so.

An interesting aside: Once a person (or any other animal) reaches terminal velocity, it "feels" to your guts and your inner ear that you aren't falling (except for the huge wind rushing past you), because you are no longer accelerating. It's the acceleration in a rollercoaster that makes your stomach turn, not the speed. This may be part of the answer to why cats who fall (accidentally) from enormous heights (more than 15 stories) have a better chance of survival than cats who fall from heights of 5 stories or so. It is hypothesized that once the cat reaches terminal velocity, it relaxes (because it no longer feels as if it's falling), and spreads out its limbs, reducing its terminal velocity still further.

--WFS 2/8/05

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