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Q & A: Balls: does size count?

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Most recent answer: 10/22/2007
Q:
Thank you very much for answering the superball question...I’ve got another question that I would really like the answer to. If two balls are made out of the same material but different masses, in theory, they should fall at the same speed as although the larger one will have more air resistance, the force of gravity will exert more force upon it as it has a larger mass and so these will both cancel out. I did an experiment at school with two superballs. They are both proportional to each other but the ball with the larger mass bounced higher. Please explain!
- Julia (age 14)
London, England
A:
Hi Julia-

Once again you are on the right track. If there were ONLY gravity around, it would exert more force on the big ball, but since the big ball has more mass it would end up accelerating exactly as much as the little ball. (That fact was discovered by Galileo, and went on to form the basis for General Relativity.) The balls would hit the floor at the same speed. Approximately, they will deform to the same shape and lose equal fractions of their energies to internal vibrations in the bounce, so they will both still bounce back at almost that same speed.

However, the friction force with the air is NOT proportional to the ball's weight or volume. At low speeds, it's proportional to the radius, and I think it becomes proportional to the area for higher speeds. Let's say the bigger ball has twice the radius, four times the area, and eight times the mass and volume of the little ball. The slowing down due to air friction goes as the force over the mass. So for the big ball it's something like two to four over eight times as large as for the small ball. Air friction slows the large ball less.

Incidentally, Galileo also wrote about how some things don't scale the same way as others- just as you've found for air friction and gravity. The amount of weight that a leg can support goes as its area, but the amount something weighs goes as its volume. If you tried to make a giant ant, its legs couldn't support its weight. The legs have to get proportionately thicker for a big animal like an elephant.

Mike W.

(published on 10/22/2007)

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