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PROBLEM 14.13
A system consists of three particles A, B, and C. We know
that 3
A
m
kg, 2
B
m
kg, and 4
C
m kg and that the
velocities of the particles expressed in m/s are, respectively,
422,
Avijk
43,
Bvij
and 242.
C vijk
Determine the angular momentum O
Hof the system about O.
O H
PROBLEM 14.14
For the system of particles of Prob. 14.13, determine (a) the
position vector r of the mass center G of the system, (b) the
linear momentum mv of the system, (c) the angular
momentum G
H of the system about G. Also verify that he
answers to this problem and to problem 14.13 satisfy the
equation given in Prob. 14.27.
SOLUTION
1.73333 1.53333 0.66667
CC
rrr i
j
k
PROBLEM 14.14 (Continued)
(c) Angular momentum about G,
2
kg m /s .
1.86667 1.46667 0.66667 0.66667 0.86667 2.33333
12 6 6 8 6 0
1.73333 1.53333 0.66667
816 8
GAAABBBCCC
mmm
Hr vr vr v
ijk ijk
ijk
PROBLEM 14.15
A 13-kg projectile is passing through the origin O with a velocity v0 (35 m/s)i when it explodes into two
fragments A and B, of mass 5 kg and 8 kg, respectively. Knowing that 3 s later the position of fragment A is
(90 m, 7 m, –14 m), determine the position of fragment B at the same instant. Assume 9.81
y
ag m/s2
and neglect air resistance.
SOLUTION
Motion of mass center:
PROBLEM 14.16
A 300-kg space vehicle traveling with a velocity v0 (360 m/s)i passes through the origin O at t 0.
Explosive charges then separate the vehicle into three parts A, B, and C, with mass, respectively, 150 kg, 100 kg,
and 50 kg. Knowing that at t 4 s, the positions of parts A and B are observed to be A (1170 m, –290 m,
–585 m) and B (1975 m, 365 m, 800 m), determine the corresponding position of part C. Neglect the effect
of gravity.
SOLUTION
Motion of mass center:
Crijk
PROBLEM 14.17
A 2-kg model rocket is launched vertically and reaches an
altitude of 70 m with a speed of 30 m/s at the end of powered
flight, time t 0. As the rocket approaches its maximum
altitude it explodes into two parts of masses 0.7 kg
A
m and
1.3 kg.
B
m Part A is observed to strike the ground 80 m west
of the launch point at t 6 s. Determine the position of part B at
that time.
SOLUTION
Choose a planar coordinate system having coordinates x and y with the origin at the launch point on the
ground and the x-axis pointing east and the y-axis vertically upward.
PROBLEM 14.18
An 18-kg cannonball and a 12-kg cannonball are chained
together and fired horizontally with a velocity of 165 m/s from
the top of a 15-m wall. The chain breaks during the flight of the
cannonballs and the 12-kg cannonball strikes the ground
at t 1.5 s, at a distance of 240 m from the foot of the wall, and
7 m to the right of the line of fire. Determine the position of the
other cannonball at that instant. Neglect the resistance of the
air.
SOLUTION
Let subscript A refer to the 12-kg cannonball and B to the 18-kg cannonball.
B
PROBLEM 14.19
Car A was traveling east at high speed
when it collided at Point O with car B,
which was traveling north at 45 mi/h. Car
C, which was traveling west at 60 mi/h,
was 32 ft east and 10 ft north of Point O
at the time of the collision. Because the
pavement was wet, the driver of car C
could not prevent his car from sliding into
the other two cars, and the three cars,
stuck together, kept sliding until they hit
the utility pole P. Knowing that the
weights of cars A, B, and C are,
respectively, 3000 lb, 2600 lb, and 2400
lb, and neglecting the forces exerted on
the cars by the wet pavement, solve the
problems indicated.
Knowing that the speed of car A was 75
mi/h and that the time elapsed from the
first collision to the stop at P was 2.4 s,
determine the coordinates of the utility
pole P.
SOLUTION
Let t be the time elapsed since the first collision. No external forces in the xy plane act on the system
consisting of cars A, B, and C during the impacts with one another. The mass center of the system moves at
the velocity it had before the collision.
P
BP
PROBLEM 14.19 (Continued)
PROBLEM 14.20
Car A was traveling east at high speed
when it collided at Point O with car B,
which was traveling north at 45 mi/h. Car
C, which was traveling west at 60 mi/h,
was 32 ft east and 10 ft north of Point O
at the time of the collision. Because the
pavement was wet, the driver of car C
could not prevent his car from sliding into
the other two cars, and the three cars,
stuck together, kept sliding until they hit
the utility pole P. Knowing that the
weights of cars A, B, and C are,
respectively, 3000 lb, 2600 lb, and 2400
lb, and neglecting the forces exerted on
the cars by the wet pavement, solve the
problems indicated. Knowing that the
coordinates of the utility pole are
46 ft
P
x and 59 ft,
P
y determine
(a) the time elapsed from the first
collision to the stop at P, (b) the speed of
car A.
SOLUTION
Let t be the time elapsed since the first collision. No external forces in the xy plane act on the system
consisting of cars A, B, and C during the impacts with one another. The mass center of the system moves at
P
BP
PROBLEM 14.20 (Continued)
A
A
PROBLEM 14.21
An expert archer demonstrates his ability by hitting tennis balls thrown by an assistant. A 2-oz tennis ball has
a velocity of (32 ft/s)i – (7 ft/s)j and is 33 ft above the ground when it is hit by a 1.2-oz arrow traveling with a
velocity of (165 ft/s)j (230 ft/s)k where j is directed upwards. Determine the position P where the ball and
arrow will hit the ground, relative to Point O located directly under the point of impact.
SOLUTION
Assume that the ball and arrow move together after the hit.
Prik
PROBLEM 14.22
Two spheres, each of mass m, can slide freely on a frictionless,
horizontal surface. Sphere A is moving at a speed 0
v 16 ft/s when it
strikes sphere B which is at rest and the impact causes sphere B to
break into two pieces, each of mass m/2. Knowing that 0.7 s after the
collision one piece reaches Point C and 0.9 s after the collision the
other piece reaches Point D, determine (a) the velocity of sphere A
after the collision, (b) the angle
and the speeds of the two pieces
after the collision.
SOLUTION
Velocities of pieces C and D after impact and fracture.
DDxDy
D
PROBLEM 14.23
In a game of pool, ball A is moving with a velocity
v
0 when it strikes
balls B and C which are at rest and aligned as shown. Knowing that after
the collision the three balls move in the directions indicated, and that
012 ft/sv
and
6.29 ft/s,
C
v
determine the magnitude of the velocity
of (a) ball A, (b) ball B.
SOLUTION
PROBLEM 14.24
A 6-kg shell moving with a velocity v0
(12 m/s) (9 m/s)
ij(360 m/s)k explodes
at Point D into three fragments A, B, and C
of mass, respectively, 3 kg, 2 kg, and 1 kg.
Knowing that the fragments hit the vertical
wall at the points indicated, determine the
speed of each fragment immediately after the
explosion. Assume that elevation changes
due to gravity may be neglected.
SOLUTION
1(34)
5
C
v
jk
PROBLEM 14.24 (Continued)
PROBLEM 14.25
A 6-kg shell moving with a velocity
v0(12 m/s) (9 m/s) (360 m/s)
ij k explodes at
Point D into three fragments A, B, and C of mass,
respectively, 2 kg, 1 kg, and 3 kg. Knowing that the
fragments hit the vertical wall at the points
indicated, determine the speed of each fragment
immediately after the explosion. Assume that
elevation changes due to gravity may be neglected.
SOLUTION
3(34)
5
C
v
jk
PROBLEM 14.25 (Continued)
72 0.70225 0.66667
AB
vv
C
PROBLEM 14.26
In a scattering experiment, an alpha particle A is projected with
the velocity 0
(600 m/s) (750 m/s) (800 m/s) uijk
into a
stream of oxygen nuclei moving with a common velocity
v0(600 m/s) .
j After colliding successively with nuclei B and
C, particle A is observed to move along the path defined by the
Points A1(280, 240, 120)and A2(360, 320, 160), while nuclei B
and C are observed to move along paths defined, respectively,
by B1(147, 220, 130), B2(114, 290, 120),and by C1(240, 232, 90)
and C2(240, 280, 75). All paths are along straight lines and all
coordinates are expressed in millimeters. Knowing that the mass
of an oxygen nucleus is four times that of an alpha particle,
determine the speed of each of the three particles after the
collisions.
PROBLEM 14.26 (Continued)
