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230 Solutions Manual
Problem 2.168
A water jet is ejected from the nozzle of a fountain with a speed
v0D12 m=s
.
Letting
ˇD33ı
, determine the rate of change of the speed of the water
particles as soon as these are ejected as well as the corresponding radius of
curvature of the water path.
Solution
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
Dynamics 2e 231
Problem 2.169
A water jet is ejected from the nozzle of a fountain with a speed
v0
. Letting
ˇD21ı
, determine
v0
so that
the radius of curvature at the highest point on the water arch is 10 ft.
Solution
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
232 Solutions Manual
Problem 2.170
A car traveling with a speed
v0D65 mph
almost loses contact with the ground when it reaches the top of
the hill. Determine the radius of curvature of the hill at its top.
Solution
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
Dynamics 2e 233
Problem 2.171
A car is traveling at a constant speed over a hill. If, using a Cartesian coordinate system with origin
O
at the top of the hill, the hill’s profile is described by the function
yD.0:003 m1/x2
, where
x
and
y
are in meters, determine the minimum speed at which the car would lose contact with the ground at
the top of the hill. Express the answer in km=h.
Solution
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
234 Solutions Manual
Problem 2.172
A race boat is traveling at a constant speed
v0D130 mph
when it performs a turn
with constant radius
⇢
to change its course by
90ı
as shown. The turn is performed
while losing speed uniformly in time so that the boat’s speed at the end of the turn is
v
fD125 mph
. If the maximum allowed normal acceleration is equal to
2g
, where
g
is the acceleration due to gravity, determine the tightest radius of curvature possible
and the time needed to complete the turn.
Solution
Let
⇢min
denote the tightest radius of curvature. The normal acceleration is
anDv2=⇢
. If the boat turns with
a constant radius of curvature
⇢D⇢min
, then the maximum normal acceleration occurs where the speed is
Dynamics 2e 235
Problem 2.173
A race boat is traveling at a constant speed
v0D130 mph
when it performs a turn
with constant radius
⇢
to change its course by
90ı
as shown. The turn is performed
while losing speed uniformly in time so that the boat’s speed at the end of the turn is
v
fD116 mph
. If the magnitude of the acceleration is not allowed to exceed
2g
, where
g
is the acceleration due to gravity, determine the tightest radius of curvature possible
and the time needed to complete the turn.
Solution
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
236 Solutions Manual
Problem 2.174
A truck enters an exit ramp with an initial speed
v0
. The ramp is a circular arc
with radius
⇢
. Derive an expression for the magnitude of the acceleration of the
truck as a function of the path coordinate
s
(and the parameters
v0
and
⇢
) if the
truck stops at
B
and travels from
A
to
B
with a constant rate of change of the
speed with respect to s.
Solution
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
Dynamics 2e 237
Problem 2.175
A jet is flying straight and level at a speed
v0D1100 km=h
when it turns to
change its course by
90ı
as shown. In an attempt to progressively tighten the
turn, the speed of the plane is uniformly decreased in time while keeping the
normal acceleration constant and equal to
8g
, where
g
is the acceleration due
to gravity. At the end of the turn, the speed of the plane is
v
fD800 km=h
.
Determine the radius of curvature
⇢
f
at the end of the turn and the time
t
f
that
the plane takes to complete its change in course.
3
3
Solution
Using normal-tangential components, the acceleration of the jet is
238 Solutions Manual
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
Dynamics 2e 239
Problem 2.176
A car is traveling over a hill with a constant speed
v0D70 mph
. Using the Cartesian coordinate system
shown, the hill’s profile is given by the function
yD.0:0005 ft1/x2
, where
x
and
y
are measured
in feet. At
xD300 ft
, the driver applies the brakes, causing a constant time rate of change of speed
PvD3ft=s2
until the car arrives at
O
. Determine the distance traveled while applying the brakes along
with the time to cover this distance. Hint: To compute the distance traveled by the car along the car’s path,
observe that ds Dpdx2Cdy2Dp1C.dy=dx/2dx, and that
Zp1CC2x2dx Dx
2p1CC2x2C1
2C ln⇣Cx Cp1CC2x2⌘:
Solution
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
