978-0073380308 Chapter 2 Solution Manual Part 21

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

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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

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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 m1/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

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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

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 ft1/x2

, where

x

and

y

are measured

in feet. At

xD300 ft

, the driver applies the brakes, causing a constant time rate of change of speed

PvD3ft=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

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permission of McGraw-Hill, is prohibited.