978-0073380308 Chapter 5 Solution Manual Part 3

870 Solutions Manual

Problem 5.21

A

30;000 lb

airplane is ﬂying on a horizontal trajectory with a speed

v0D650 mph

when, at point

A

, it maneuvers so that at point

B

it is on a

steady climb with

✓D40ı

and a speed of

600 mph

. If the change in mass

of the plane between

A

and

B

is negligible, determine the impulse that

had to be exerted on the plane in going from Ato B.

NAVY

3

Solution

NAVY

3

We model the airplane as a particle. We denote by

E

FT

the total force acting on the airplane.

We denote by

t1

and

t2

the time instants at which the airplane is at

A

and

B

, respectively. We

use subscripts

1

and

2

to denote quantities at

t1

and

t2

, respectively. Since we are provided

with information that completely identifies the velocity of the airplane at times

t1

and

t2

,

we will be able to determine the inputs acting on the airplane by a direct application of the

impulse-momentum principle.

Balance Principles. Applying the impulse momentum principle, we have

mEv1CZt2

t1E

FTdt DmEv2;(1)

where mand Evare the mass and velocity of the airplane, respectively.

Force Laws. All forces are accounted for on the FBD.

Kinematic Equations. The velocities at times t1and t2are

Ev1Dv0O{and Ev2Dv2.cos ✓O{Csin ✓O|/; (2)

where v2D600 mph.

Computation.

The integral on the left-hand side of Eq. (1) is the impulse of the force

E

FT

. Hence, we have

Zt2

t1E

FTdt Dm.Ev2Ev1/: (3)

Substituting Eqs. (2) into Eq. (3), we then have

Zt2

t1E

FTdt Dm⇥.v2cos ✓v0/O{Cv2sin ✓O|⇤:(4)

Recalling that

mD30;000 lb=g

,

gD32:2 ft=s2

,

v2D600 mph D6005280

3600 ft=s

,

✓D40ı

, and

v0D

650 mph D6505280

3600 ft=s

, we can evaluate the impulse that acted on the airplane between

t1

and

t2

to obtain

Zt2

t1E

FTdt D.260:1⇥103O{C527:0⇥103O|/lbs:

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June 25, 2012

Dynamics 2e 871

Problem 5.22

A crate starts sliding from rest down an incline with

✓D35ı

. Determine the speed

of the crate after

2:5

s if the coefficient of kinetic friction between the crate and the

incline is kD0:25. Express the result in SI units.

Solution

We model the crate as a particle subject only to its own weight

mg

and the force due

to the contact with the incline, where the force in question has been represented via

872 Solutions Manual

Problem 5.23

A trebuchet launches a projectile with an initial velocity

Ev0

such that the

projectile takes

3

s to achieve its maximum height with a corresponding speed

of 145 ft=s. Use the impulse-momentum principle to determine Ev0.

Solution

We model the projectile as a particle subject only to its own weight

mg

. For convenience we

denote by

t1D0

the time instant at which the projectile becomes airborne. In addition we

Dynamics 2e 873

Problem 5.24

A

15 oz

football is kicked straight up in the air such that it takes

2:7

s (after

separating from the kicker’s foot) to achieve its maximum height. Determine

the impulse imparted by the kicker to the football, assuming that right before

the kick the football is held stationary and that the weight of the football can be

neglected while it is in contact with the foot. Also, if the contact between the

football and the foot lasts

8⇥103

s, determine the average force exerted by the

kicker on the football.

Solution

We begin by defining three time instant:

t1D0

is the time when

the ball first comes into contact with the foot;

t2D8⇥103

s is the

874 Solutions Manual

E

of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the

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Dynamics 2e 875

Problem 5.25

The takeoff runway on carriers is much too short for a modern jetplane to take off on its own. For this

reason, the takeoff of carrier planes is assisted by hydraulic catapults (Fig. A). The catapult system is

housed below the deck except for a relatively small shuttle that slides along a rail in the middle of the

runway (Fig. B). The front landing gear of carrier planes is equipped with a tow bar that, at takeoff, is

attached to the catapult shuttle (Fig. C). When the catapult is activated, the shuttle pulls the airplane along

the runway and helps the plane reach its takeoff speed. The takeoff runway is approximately

300 ft

long,

and most modern carriers have three or four catapults.

In a catapult-assisted takeoff, assume that a

45;000 lb

plane goes from

0

to

165 mph

in

2

s while

traveling along a rectilinear and horizontal trajectory. Also assume that throughout the takeoff the plane’s

engines are providing 32;000 lb of thrust.

(a) Determine the average force exerted by the catapult on the plane.

(b)

Now suppose that the takeoff order is changed so that a small trainer aircraft must take off first. If

the trainer’s weight and thrust are 13,000 and

5850 lb

, respectively, and if the catapult is not reset to

match the takeoff specifications for the smaller aircraft, estimate the average acceleration to which the

trainer’s pilots would be subjected and express the answer in terms of

g

. What do you think would

happen to the trainer’s pilot?

Photo credit (A): U.S. Navy photo by Photographer’s Mate 2nd Class H. Dwain Willis

Photo credit (B): PHAN James Farrally II, U.S. Navy

Photo credit (C): U.S. Navy photo by Photographer’s Mate 3rd Class (AW) J. Scott Campbell

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.

876 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 877

Problem 5.26

The takeoff runway on carriers is much too short for a modern jetplane to take off on its own. For this

reason, the takeoff of carrier planes is assisted by hydraulic catapults (Fig. A). The catapult system is

housed below the deck except for a relatively small shuttle that slides along a rail in the middle of the

runway (Fig. B). The front landing gear of carrier planes is equipped with a tow bar that, at takeoff, is

attached to the catapult shuttle (Fig. C). When the catapult is activated, the shuttle pulls the airplane along

the runway and helps the plane reach its takeoff speed. The takeoff runway is approximately

300 ft

long,

and most modern carriers have three or four catapults.

If the carrier takeoff of a

45;000 lb

plane subject to the

32;000 lb

thrust of its engines were not assisted

by a catapult, estimate how long it would take for a plane to safely take off, i.e., to reach a speed of

165 mph starting from rest. Also, how long a runway would be needed under these conditions?

Photo credit (A): U.S. Navy photo by Photographer’s Mate 2nd Class H. Dwain Willis

Photo credit (B): PHAN James Farrally II, U.S. Navy

Photo credit (C): U.S. Navy photo by Photographer’s Mate 3rd Class (AW) J. Scott Campbell

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.

878 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 879

Problem 5.27

A

10 lb

box is released from rest at a distance

dD5ft

from the bottom of a smooth chute with

✓D30ı

.

The friction between the chute and the box is negligible, and so is the change in speed of the box in going

from the inclined to the horizontal part of the chute. If the duration of the transition between inclined and

horizontal motion takes 0:02 s, determine the average force acting on the box during this transition.

Solution

We begin by defining three time instant:

t1D0

is the time when the box starts

moving;

t2

is the time at which the box arrives at the bottom of the incline;