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In the previous problem, we demonstrate that the force of air resistance on a sphere of radius R can plausibly be argued to have the form

where the v with an arrow is the vector velocity and the v without is its magnitude (the speed). The density of the air, r, is about 1 kg/m^{3} --- 1/1000 that of water. The parameter C is a dimensionless constant.

If we drop a steel ball and a styrofoam ball from a height of *s*, the steel ball reaches the ground when the styrofoam ball is still a bit above the ground -- call the distance *h*. Estimate the air resistance coefficient C as follows:

(b) Since the steel and styrofoam were not very different, use *v*_{av}, the average velocity of the steel ball during its fall to calculate an average air resistance force, *F*_{av} = -b v_{av}^{2},
acting on the styrofoam sphere during its fall. Express this force in terms of *b*, *m* (the mass of the styrofoam sphere), g, *s*, and *h*.

(c) The average velocity of the steel ball is *v*_{av,steel} = *s*/*t*. The average velocity of the styrofoam sphere was *v*_{av,sty }= (*s-h*)/*t*. Assume this difference is caused by the average air resistance force acting over the time *t* with our basic Newton's law formula:

Use this to show that

(d) A styrofoam ball of radius *R* = 5 cm and mass *m* = 50 gm, is dropped with a steel ball from a height of *s* = 2 m. When the steel ball hits, the styrofoam is about *h* = 10 cm above the ground. Calculate b (for the styrofoam sphere) and C (for any sphere).

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Page last modified October 15, 2002: D29