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900 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 901
Problem 5.40
Two canoeists
A
and
B
are drifting downstream with a common
speed
v0D8m=s
. At some point,
A
and
B
use a rope to reduce
the distance between them. If
A
and
B
can reduce their distance
at a rate of
1m=s
, determine the velocity of
A
and
B
when they
finally come together. Let the masses of
A
and
B
(including their
respective canoes) be
mAD90 kg
and
mBD75 kg
, respectively.
In addition, neglect the drag acting on the canoes due to the water.
C01161963 C01161963
Solution
C01161963
C01161963
We model both
A
and
B
as a system of particles. Referring to the figure
at the right, we assume that
A
and
B
, as a system, are subject only to
their own weights
mAg
and
mBg
, respectively, and to the buoyancy
forces
NA
and
NB
, acting on
A
and
B
, respectively. For convenience,
we denote by
t1
the time instant before the distance between
A
and
B
is
reduced, and we denote by
t2
a time instant right before
A
and
B
come
together. We use subscripts 1and 2to denote quantities at t1and t2, respectively.
Balance Principles.
Since there are no external forces in the
x
direction, the linear momentum of the
system must be conserved in that direction. So,
mA.vAx/1CmB.vBx/1DmA.vAx /2CmB.vBx/2;(1)
where vAx and vBx are the velocity components of Aand B, respectively.
Force Laws. All forces have been accounted for in the FBD.
Kinematic Equations.
At time
t1
,
A
and
B
are moving to the right with the same speed
v0
. Once
A
and
B
start reducing their distance, their relative velocity depends on the specified rate with which
A
and
B
diminish their distance. Hence, summarizing, we have
.vAx/1Dv0; .vBx/1Dv0;and .vBx/2.vAx/2Dvr;(2)
where vrdenotes the magnitude of the rate with which the distance between Aand Bdecreases.
Computation.
The equation resulting from the substitution of the first two of Eqs. (2) into Eq. (1) along
with the last of Eqs. (2) form the following system of two equations in the two unknowns
.vAx/2
and
.vBx/2
:
.mACmB/v0DmA.vAx/2CmB.vBx/2and .vBx/2.vAx/2Dvr;(3)
whose solution is
.vAx/2Dv0CmBvr
mACmB
and .vBx/2Dv0mAvr
mACmB
:(4)
Recalling that
EvADvAx O{
,
EvBDvBx O{
,
mAD90 kg
,
mBD75 kg
,
v0D8m=s
, and
vrD1m=s
, using
Eqs. (4) we can evaluate the velocities of Aand Bright before they come together to obtain
EvAD8:455 O{m=s and EvBD7:455 O{m=s:
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
902 Solutions Manual
Problem 5.41
Suppose that a
180 lb
person
A
(the weight includes the the rifle and ammu-
nition before firing) were to fire a
180 gr
(
7000 gr D1lb
) bullet
B
with a
muzzle velocity of 3300 ft=s. If the person fires while resting on ice, so that
the friction between the person and the ground is negligible, determine the
final velocity of both the person and the bullet.
Solution
We model both
A
and
B
as a system of particles. Referring to the
figure at the right, we assume that
A
and
B
, as a system, are subject
Dynamics 2e 903
Problem 5.42
Blocks
A
and
B
, with masses
mAD400 kg
and
mBD90 kg
, respectively, are initially at rest when block
Astarts sliding down the incline with ✓D30ı.
If friction between the two blocks and between block
A
and the incline is negligible, use the impulse-
momentum principle to determine the velocities of Aand B 1:5 s after release.
Solution
We model blocks
A
and
B
as particles subject to their own weights
mAg
and
mBg
,
respectively, and to the normal forces
NAB
and
NA
due to the contact between
A
and
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 905
Problem 5.43
Blocks
A
and
B
, with masses
mAD400 kg
and
mBD90 kg
, respectively, are initially at rest when block
Astarts sliding down the incline with ✓D30ı.
Let
kD0:15
be the kinetic coefficient of friction between block
A
and the incline. If friction between
the two blocks is negligible, determine the velocities of Aand B 1:5 s after release.
Solution
We model blocks
A
and
B
as particles subject to their own weights
mAg
and
mBg
,
respectively, (
mA
is the mass of
A
,
mB
is the mass of
B
, and
g
is the acceleration
906 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 907
Problem 5.44
Two persons
A
and
B
weighing 140 and
180 lb
, respectively, jump
off a floating platform (in the same direction) with a velocity relative
to the platform that is completely horizontal and with magnitude
v0D6ft=s
for both
A
and
B
. The floating platform weighs
800 lb
.
Assume that A,B, and the platform are initially at rest.
Neglecting the water resistance to the horizontal motion of the
platform, determine the speed of the platform after
A
and
B
jump
at the same time.
Solution
Referring to the FBD on the right and denoting the platform by
P
, we model
A
,
B
,
and
P
as a system, of particles subject to their respective weights
mAg
,
mBg
, and
908 Solutions Manual
Problem 5.45
Two persons
A
and
B
weighing 140 and
180 lb
, respectively, jump
off a floating platform (in the same direction) with a velocity relative
to the platform that is completely horizontal and with magnitude
v0D6ft=s
for both
A
and
B
. The floating platform weighs
800 lb
.
Assume that A,B, and the platform are initially at rest.
Neglecting the water resistance to the horizontal motion of the
platform, and knowing that
B
jumps first, determine the speed of
the platform after both Aand Bhave jumped.
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 909
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.
