978-0073380308 Chapter 2 Solution Manual Part 27

290 Solutions Manual

Problem 2.218

Reference frame

A

is translating relative to reference frame

B

with velocity

EvA=B

and acceleration

EaA=B

.

A particle

C

appears to be stationary relative to frame

A

. What can you say about the velocity and

acceleration of particle Crelative to frame B?

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 291

Problem 2.219

Reference frame

A

is translating relative to reference frame

B

with constant velocity

EvA=B

. A particle

C

appears to be in uniform rectilinear motion relative to frame

A

. What can you say about the motion of

particle Crelative to frame B?

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.

292 Solutions Manual

Problem 2.220

A skier is going down a slope with moguls. Let the skis be short enough for

us to assume that the skier’s feet are tracking the moguls’ profile. Then, if the

skier is skilled enough to maintain her hips on a straight-line trajectory and

vertically aligned over her feet, determine the velocity and acceleration of her

hips relative to her feet when her speed is equal to

15 km=h

. For the profile

of the moguls, use the formula

y.x/ DhI0:15x C0:125 sin.⇡x=2/ m

,

where hIis the elevation at which the skier starts the descent.

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 293

Problem 2.221

Two particles

A

and

B

are moving in a plane with arbitrary velocity vectors

EvA

and

EvB

, respectively.

Letting the rate of separation (ROS) be defined as the component of the relative velocity vector along the

line connecting particles

A

and

B

, determine a general expression for

ROS

. Express your result in terms

of

ErB=A DErBErA

, where

ErA

and

ErB

are the position vectors of

A

and

B

, respectively, relative to some

chosen fixed point in the plane of motion.

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.

294 Solutions Manual

Problem 2.222

Airplanes

A

and

B

are ﬂying along straight lines at the same altitude and with speeds

vAD660 km=h

and

vBD550 km=h, respectively. Determine the speed of Aas perceived by Bif ✓D50ı.

3

Solution

3

Referring to the figure at the right, we use the Cartesian coordinate system

shown to express the velocities of the airplanes Aand B:

EvADvAO|and EvBDvB.sin ✓O{Ccos ✓O|/: (1)

Using relative kinematics, the velocity of Aas perceived by Bis

EvA=B DEvAEvBDvBsin ✓O{C.vAvBcos ✓/O|: (2)

The speed of

A

as perceived by

B

is the magnitude of the vector

EvA=B

. Using

the result in Eq. (2), we have

vA=B DˇˇEvA=B ˇˇDqv2

Bsin2✓C.vAvBcos ✓/2:(3)

Recalling that

vAD660 km=hD6601000

3600 m=s

,

vBD550 km=hD

5501000

3600 m=s and ✓D50ı, we can evaluate the expression in Eq. (3) to obtain

vA=B D144:7 m=s:

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 295

Problem 2.223

Three vehicles

A

,

B

, and

C

are in the positions shown and are

moving with the indicated directions. We define the rate of sepa-

ration (ROS) of two particles

P1

and

P2

as the component of the

relative velocity of, say,

P2

with respect to

P1

in the direction of

the relative position vector of

P2

with respect to

P1

, which is along

the line that connects the two particles. At the given instant, de-

termine the rates of separation

ROSAB

and

ROSCB

, that is, the

rate of separation between

A

and

B

and between

C

and

B

. Let

vAD60 mph

,

vBD55 mph

, and

vCD35 mph

. Furthermore, treat

the vehicles as particles and use the dimensions shown in the figure.

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.

296 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 297

Problem 2.224

Car

A

is moving at a constant speed

vAD75 km=h

, while car

C

is

moving at a constant speed

vCD42 km=h

on a circular exit ramp

with radius

⇢D80

m. Determine the velocity and acceleration of

C

relative to A.

Solution

Referring to the figure at the right, the

xy

frame of reference with origin

298 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 299

Problem 2.225

During practice, a player

P

punts a ball

B

with a speed

v0D25 ft=s

,

at an angle

✓D60ı

, and at a height

h

from the ground. Then the

player sprints along a straight line and catches the ball at the same height

from the ground at which the ball was initially kicked. The length

d

denotes the horizontal distance between the player’s position at the start of

the sprint and the ball’s position when the ball leaves the player’s foot.

Also, let

t

denote the time interval between the instant at which the

ball leaves the player’s foot and the instant at which the player starts

sprinting.

Assume that

dD0

and

tD0

, and determine the average speed of

the player so that he catches the ball.

Solution