THE UNIVERSITY OF BRITISH COLUMBIA
 
Science 1 Physics Assignment # 2:
 
Calculus and Kinematics
 
Thurs. 21 Sept. 2000 - finish by Thurs. 28 Sept. 2000

1.

Free Fall with Friction
For very low velocities (before turbulence sets in) the drag force D on an object moving through a fluid [e.g. air] is proportional to the velocity v of the object and (as always, for friction) directed opposite to the motion:

$$D \; = \; - b \; v$$

where  b  is some constant with units of force per unit velocity. If the object is falling under the influence of a constant gravitational force  mg  in the downward direction (which we arbitrarily pick as the positive direction), we can write NEWTON'S SECOND LAW (F = m a) explicitly as   $m \; a \; = \; m \; g \; - b \; v$. Dividing through by  m  and noting that   $a \equiv dv / dt$,  we get

$${dv \over dt} \; = \; g \; - k \; v \qquad \hbox{\rm where} \qquad k \equiv {b \over m}.$$

(a)

Terminal velocity: At what  v_f  will the velocity no longer change with time?
(Explain your answer.)

(b)

Change of variable: Find some function of  v  called  u  for which   ${\displaystyle {du \over dt} \; = - k \; u }$.

(c)

Time dependence: Find the velocity  v  as an explicit function of time, assuming that the falling object starts at rest (v=0 at t=0).

. . . and Tipler Ch. 2:

(4$^{\rm th}$ Edition) problems 6, 20, 57, 91 and 127 -- or -- (3$^{\rm rd}$ Edition) problems 14, 29, 37 and 71

. . . and Tipler Ch. 3:

(4$^{\rm th}$ Edition) problems 93, 105 and 113 -- or -- (3$^{\rm rd}$ Edition) problems 63 and 66