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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
3
Solution
NAVY
3
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.Ev2Ev1/: (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|/lbs:
This solutions manual, in any print or electronic form, remains the property of McGraw-Hill, Inc. It may be used and/or possessed only by permission
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.
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⇥103
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⇥103
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
permission of McGraw-Hill, is prohibited.
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;
