Page 59 - Advanced engineering mathematics

1.5 Additional Applications 39

SECTION 1.5 PROBLEMS

1. A 10-pound ballast bag is dropped from a hot air bal- 10 Ω
loon which is at an altitude of 342 feet and ascending
at 4 feet per second. Assuming that air resistance is
not a factor, determine the maximum height reached
by the bag, how long it remains aloft, and the speed
with which it eventually strikes the ground.
2. A 48 pound box is given an initial push of 16 feet 15 Ω
per second down an inclined plane that has a gradi- 30 Ω
10 V
ent of 7/24. If there is a coefficient of friction of 1/3
between the box and the plane and a force of air resis-
tance equal in magnitude to 3/2 of the velocity of the
box, determine how far the box will travel down the
plane before coming to rest.
FIGURE 1.13 Circuit of Problem 8, Section 1.5.
3. A skydiver and her equipment together weigh 192
pounds. Before the parachute is opened, there is an air the capacitor voltage be 76 volts? Determine the cur-
drag force equal in magnitude to six times her veloc- rent in the resistor at that time. The resistances are
ity. Four seconds after stepping from the plane, the in thousands of ohms, and the capacitor is in micro-
skydiver opens the parachute, producing a drag equal farads (10 −6 farads).
to three times the square of the velocity. Determine the
velocity and how far the skydiver has fallen at time t.
What is the terminal velocity?
250
4. Archimedes’ principle of buoyancy states that an
object submerged in a fluid is buoyed up by a force
equal to the weight of the fluid that is displaced by
2
the object. A rectangular box of 1 × 2 × 3 feet and
weighing 384 pounds is dropped into a 100-foot deep 80 V
freshwater lake. The box begins to sink with a drag
due to the water having a magnitude equal to 1/2the
velocity. Calculate the terminal velocity of the box.
Will the box have achieved a velocity of 10 feet per
second by the time it reaches the bottom? Assume that
FIGURE 1.14 Circuit of Problem 9, Section 1.5.
the density of water is 62.5 pounds per cubic foot.
5. Suppose the box in Problem 4 cracks open upon hit- 10. For the circuit in Figure 1.15, find all currents imme-
ting the bottom of the lake, and 32 pounds of its con- diately after the switch is closed, assuming that
tents spill out. Approximate the velocity with which all of these currents and the charges on the capaci-
the box surfaces. tors are zero just prior to closing the switch. Resis-
6. The acceleration due to gravity inside the earth is pro- tances are in ohms, the capacitor in farads, and the
portional to the distance from the center of the earth. inductor in henrys.
An object is dropped from the surface of the earth into 10 Ω 30 Ω 15 Ω
a hole extending straight through the planet’s center.
Calculate the speed the object achieves by the time it
reaches the center.
7. A particle starts from rest at the highest point of a
vertical circle and slides under only the influence of
5 f
gravity along a chord to another point on the circle.
1/10 h 4/10 h
Show that the time taken is independent of the choice
of the terminal point. What is this common time?
6 V
8. Determine the currents in the circuit of Figure 1.13.
9. In the circuit of Figure 1.14, the capacitor is initially
discharged. How long after the switch is closed will FIGURE 1.15 Circuit of Problem 10, Section 1.5.

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