WW07: Problem 22
(1 point)
(????) Imagine dropping a steel ball bearing into a tall cylinder full of glycerine. Viscous drag has a significant effect on the ball bearing's
downward velocity, v(t). One family of dynamic models for this takes the form
(*)
dv
dt
= g - kv<sup>p</sup>,
where g is the acceleration of gravity and k is a constant. The size of k is related to the viscosity of the medium: when k = 0, equation (*) is the
familiar friction-free approximation for a falling mass. The exponent p is an interesting topic for dicussion: for this problem we consider p = 1.
In this problem, we use SI units, so t is in seconds, v is in metres per second, etc. Use the international standard value g = 9.807 and assume
glycerine has viscosity parameter k = 23.5.
(a) Find v(t). (Assume v(0) = 0.)
v(t) =
(b) Let y(t) denote the distance the ball bearing has dropped at time t. (This terminology entails y(0) = 0, and y(t) > 0 for t > 0.) Find y(t).
y(t) =
(c) Using computer assistance as appropriate, find the time required for the ball bearing to drop 1 metre from its release point.
y(t) = 1 when t =
(Enter at least 4 significant figures.)
