James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

applied project Which Is Faster, Going Up or Coming Down? 609
tee that the water pressure will be at least 2160 lbyft 2 for a period of 10 minutes. (When
a fire happens, the electrical system might fail and it could take up to 10 minutes for the
emergency generator and fire pump to be activated.) What height should the engineer
specify for the tank in order to make such a guarantee? (Use the fact that the water pressure
at a depth of d feet is P − 62.5d. See Section 8.3.)
4. Not all water tanks are shaped like cylinders. Suppose a tank has cross-sectional area Ashd
at height h. Then the volume of water up to height h is V − y h Asud du and so the Funda -
0
mental Theorem of Calculus gives dVydh − Ashd. It follows that
and so Torricelli’s Law becomes
dV
dt
− dV
dh
Ashd dh
dt
dh
dt
− Ashd dh
dt
− 2as2th
(a) Suppose the tank has the shape of a sphere with radius 2 m and is initially half full of
water. If the radius of the circular hole is 1 cm and we take t − 10 mys 2 , show that h
satisfies the differential equation
s4h 2 h 2 d dh
dt
− 20.0001s20h
(b) How long will it take for the water to drain completely?
applied Project
In modeling force due to air resistance,
various functions have been
used, depending on the physical
characteristics and speed of the ball.
Here we use a linear model, 2pv,
but a quadratic model (2pv 2 on the
way up and pv 2 on the way down)
is another possibility for higher
speeds (see Exercise 9.3.50). For a
golf ball, experiments have shown
that a good model is 2pv 1.3 going
up and p| v | 1.3 coming down. But no
matter which force func tion 2f svd
is used [where f svd . 0 for v . 0
and f svd , 0 for v , 0], the answer
to the question remains the same.
See F. Brauer, “What Goes Up Must
Come Down, Eventually,” American
Mathematical Monthly 108 (2001),
pp. 437–440.
which is faster, going up or coming down?
Suppose you throw a ball into the air. Do you think it takes longer to reach its maximum height
or to fall back to earth from its maximum height? We will solve the problem in this project,
but before getting started, think about that situation and make a guess based on your physical
intuition.
1. A ball with mass m is projected vertically upward from the earth’s surface with a positive
initial velocity v 0. We assume the forces acting on the ball are the force of gravity and a
retarding force of air resistance with direction opposite to the direction of motion and with
magnitude p| vstd |, where p is a positive constant and vstd is the velocity of the ball at time t.
In both the ascent and the descent, the total force acting on the ball is 2pv 2 mt. [During
ascent, vstd is positive and the resistance acts downward; during descent, vstd is negative and
the resistance acts upward.] So, by Newton’s Second Law, the equation of motion is
mv9 − 2pv 2 mt
Solve this differential equation to show that the velocity is
vstd −Sv 0 1 mt
De 2ptym 2 mt
p p
2. Show that the height of the ball, until it hits the ground, is
ystd −Sv 0 1 D mt m p p s1 2 e2ptym d 2 mtt
p
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