Picture an object in free fall. If the leading face of the object (which means the face closest to the ground) has a large surface area, air resistance becomes important. For low speeds, the force due to air resistance may be approximated as proportional to the object's velocity according to the expression \(\vec{F}^{a}=-b \vec{v}\), where \(b\) is a number that depends on the shape and volume of the object and on the density of the air.

(a) Show that, for an object of inertia \(m\), there is a limiting speed (called the terminal speed) given by \(v_{\mathrm{t}}=m g / b\).

(b) Show that, if the object is released from rest, its speed as a function of time is given by

(c) What are the units of \(m / b\) ? Sketch a graph of \(v\) as a function of time, with the horizontal axis in units of multiples of \(m / b\).

Transcribed Image Text:

v(t) =[1-(b/mt].