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)
