Question

A helicopter hovers 500 feet above ground, over a large open tank full of liquid (not water). A dense compact object weighing 160 pounds is dropped (released from rest) from the helicopter into the liquid. Assume that air resistance is proportional to instantaneous velocity v while the object is in the air and that viscous damping is proportional to v2 after the object has entered the liquid. For air take the constant of proportionality to be k = 1 /A helicopter hovers 500 feet above ground, over a large open tank full of liquid (not water). A dense compact object weighing 160 pounds is dropped (released from rest) from the helicopter into the liquid. Assume that air resistance is proportional to instantaneous velocity v while the object is in the air and that viscous damping is proportional to

v2

after the object has entered the liquid. For air take the constant of proportionality to be

k =

1

4

,

and for the liquid take it to be

k = 0.1.

Assume that the positive direction is downward. If the (above-ground) tank is 75 feet high, determine the time and the impact velocity when the object hits the bottom of the tank. [Hint: Think in terms of two distinct IVPs. If you use (13) from Section 3.2,

du

a2 − u2

=

1

2a

ln

a + u

a − u

+ c,    |u| ≠ a,

be careful in removing the absolute value sign. You might compare the velocity when the object hits the liquid—the initial velocity for the second problem—with the terminal velocity

vt

of the object falling through the liquid.] (Assume the acceleration due to gravity is

g = 32 ft/s2,

and the mass is m = 160/g. Round your answers to five decimal places.)

time     s

velocity     ft/s

4 , and for the liquid take it to be k = 0.1. Assume that the positive direction is downward. If the (above-ground) tank is 75 feet high, determine the time and the impact velocity when the object hits the bottom of the tank. [Hint: Think in terms of two distinct IVPs. If you use (13) from Section 3.2,

du a2 − u2 = 1 2a ln a + u a − u + c, |u| ≠ a, be careful in removing the absolute value sign. You might compare the velocity when the object hits the liquid—the initial velocity for the second problem—with the terminal velocity vt of the object falling through the liquid.] (Assume the acceleration due to gravity is g = 32 ft/s2, and the mass is m = 160/g. Round your answers to five decimal places.) time s velocity ft/s

Answer 1