MadSci Network: Physics
Query:

Re: Will 2 falling objects of diiferent mass hit at the same time?

Date: Tue Apr 7 14:11:50 1998
Posted By: Eric Kramer, post-doc, physics/chemistry, Brandeis University
Area of science: Physics
ID: 891689338.Ph
Message:

Hmm. It sounds to me like your coworkers are correct.

Let's consider the contributions to Newton's law for a 
metal ball of radius R and mass M, falling through the air.
The force due to gravity is

F_gravity = M*g

where g is the acceleration due to gravity in a vacuum (g = 9.80 m/s^2).
Since the density of metals is typically more than 1000 times 
greater than the density of air, we can neglect the boyancy force here. 
However, boyancy will be important for a styrofoam ball.

The force due to air resistance is more difficult. It depends
on the velocity and the radius of the ball in a complicated way.
However, at low speeds it is approximately linear in the velocity V and 
the radius R:

F_air_resistance = -A*V*R 

where A = 0.00037 kg/(m*s)
and the negative sign indicates that the force opposes the motion.

The acceleration of the ball can be found using Newton's law:

acceleration = (sum of forces)/(mass) 
             = (M*g - A*V*R)/M
             =  g - A*V*R/M

While the mass cancels out of the term due to gravity, it ends
up in the denominator of the term due to air resistance.
Other things being equal, a less massive ball will fall slower!

This is generally true even at high speeds, where the force
of air resistance increases like the *square* of the velocity.



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