Question

We learned in mechanics that in the absence of air resistance, all objects near Earth’s surface fall with the same constant acceleration g = 9.8 m/s 2 . Yet as a student I once dropped a wadded-up tissue from the top of the Eiffel Tower and watched it take minutes to fall 300 m to Earth. (I retrieved it from the riverbank and put it in the poubelle.) In fluids, such as air or water, frictional drag forces grow larger as speed increases: often Fdrag ∝ v (or Fdrag ∝ v 2 if turbulence is present). The result is that once a falling object reaches terminal velocity, where the upward drag force balances the downward gravitational force, the object no longer accelerates: it continues downward at constant speed. (a) If an object falling downward at speed v feels an upward drag force Fdrag = bv (where b is some constant that depends on the object’s size and shape and the fluid properties), what is its terminal speed vterm? (Write vterm in terms of m, g, and b.) (b) Using your expression for vterm, explain how it makes sense that a ping-pong ball dropped from a rooftop takes longer to fall to Earth than a steel ball of the same size and shape.

Answer 1

The terminal speed will be attained when the net force on the object becomes zero i.e the downward force of weight is balanced by the upward drag force.

b) As can be seen from above relation terminal speed is directly proportional to the mass of the object. Since the mass of steel ball is more than ping- pong ball of the same size the terminal speed of steel ball is more and hence it takes less time to cover the distance and ping - pong ball has less terminal speed so it takes more time to cover the same distance.