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y -
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'.
,.~ ,
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64
Chapter 2. First Order Differential Equations
TABLE 2.3.2 Volume and Flow Data for the Great
Lakes
I
,
I
~
I~:
~
Lake
V (km3 x 103)
r (km3/year)
Superior
Michigan
Erie
Ontario
12.2
4.9
0.46
1.6
65.2
158
175
209
25. A body of constant mass m is proj
medium offering a resistance klvl,
tional force.
(a) Find the maximum height Xm
maximum height is reached.
(b) Show that if kvo/mg < 1, theIJ
20. A ball with mass 0.15 kg is thrown upward with initial velocity 20 mlsec from the roof of
a building 30 m high. Neglect air resistance.
(a) Find the maximum height above the ground that the ball reaches.
(b) Assuming that the ball misses the building on the way down, find the time that it hits
the ground.
(c) Plot the graphs of velocity and position versus time.
21. Assume that the conditions are as in Problem 20 except that there is a force due to air
resistance of Ivl/30, where the velocity v is measured in m/sec.
(a) Find the maximum height above the ground that the ball reaches.
(b) Find the time that the ball hits the ground.
(c) Plot the graphs of velocity and position versus time. Compare these graphs with the
corresponding ones in Problem 20.
~
~
,I' :'""
Iii
ll
2.3 Modeling with First Order Equations
22. Assume that the conditions are as in Problem 20 except that there is a force due to air
resistance of v2/1325, where the velocity v is measured in mlsec.
(a) Find the maximum height above the ground that the ball reaches.
(b) Find the time that the ball hits the ground.
(c) Plot the graphs of velocity and position versus time. Compare these graphs with the
corresponding ones in Problems 20 and 21.
(c) Show that the quantity kvo/m~
26. A body of mass m is projected veJ
offering a resistance kJvl, where k
of the earth is constant.
(a) Find the velocity v(t) of the be
(b) Use the result of part (a) to cal4
approaches zero. Does this result
with an initial velocity Voin a vacw
(c) Use the result of part (a) to c~
approaches zero.
27. A body falling in a relatively dense:
Figure 2.3.5): a resistive force R, a
buoyant force is equal to the weight
spherical body of radius a, the resis
v is the velocity of the body, and J1
23. A sky diver weighing 180 lb (including equipment) falls vertically downward from an
altitude of 5000 ft and opens the parachute after 10 sec of free fall. Assume that the force
of air resistance is 0.751vl when the parachute is closed and 121vlwhen the parachute is
open, where the velocity v is measured in ftIsec.
(a) Find the speed of the sky diver when the parachute opens.
(b) Find the distance fallen before the parachute opens.
(c) What is the limiting velocity VLafter the parachute opens?
(d) Determine how long the sky diver is in the air after the parachute opens.
(e) Plot the graph of velocity versus time from the beginning of the fall until the skydiver
reaches the ground.
24. A rocket sled having an initial speed of 150 mi/hr is slowed by a channel of water. Assume
that, during the braking process, the acceleration a is given by a(v) = _J.w2,where v is the
velocity and J1.is a constant.
( ~ '\
A ~ in P v~mn1p
4 1. " th p. tp.yt
11~P. th P: rp.l~hnn.
rlu/dt
,,(d,,/dr)
to write
.
the eClIIMion
FIGURE 2.3.5
65
2.3 Modeling with First Order Equations
--~
,Jer[)ifferential Equations
---
r Ih~Great
km 'year)
65.2
158
1'5
209
25. A body of constant mass m is projected vertically upward with an initial velocity Voin a
medium offering a resistance klvl, where k is a constant. Neglect changes in the gravitational force.
(a) Find the maximum height Xm attained by the body and the time tm at which this
maximum height is reached.
(b) Show that if kvo/mg
< 1, then tm and Xm can be expressed
= Vo 1- ~ kvo + ~
g [
2 mg 3
kVO
(mg ) -..
-..
Xm= v~ 1 - ~kvo + ~
2g [
3 mg 2 (mg )
tm
kVO
locity20 mlsec from the roof of
ball reaches.
v down,find the time that it hits
thaIthereis a force due to air
"sec,
ball reaches.
Compare these graphs with the
Ihallhere is a force due to air
m/sec.
hall reaches.
( ornpare these graphs with the
2
. ,
J
. .
J
(c) Show that the quantity kvo/mg is dimensionless.
26. A body of mass m is projected vertically upward with an initial velocity Voin a medium
offering a resistance klvl, where k is a constant. Assume that the gravitational attraction
of the earth is constant.
(a) Find the velocity v(t) of the body at any time.
(b) Use the result of part (a) to calculate the limit of v(t) as k -+ 0, that is, as the resistance
approaches zero. Does this result agree with the velocityof a massm projected upward
with an initial velocity Voin a vacuum?
(c) Use the result of part (a) to calculate the limit of v(t) as m -+ 0, that is, as the mass
approaches zero.
27. A body falling in a relatively dense fluid, oil for example, is acted on by three forces (see
Figure 2.3.5): a resistive force R, a buoyant force B, and its weight w due to gravity. The
buoyantforceisequal to the weightofthe fluiddisplacedbythe object. Foraslowlymoving
spherical body of radius a, the resistive force is given by Stokes'? law, R = 6Jl'J,talvl, where
v is the velocity of the body, and J,tis the coefficient of viscosity of the surrounding fluid.
Verticallydownward from an
freefall.A.ssurnethat the forc.e
nd 121vI
when the parachute IS
ParachUte opens.
,i,".
'''' l~e fall Until the sky d'1ver
2
as
Rlr
'J