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PROBLEM 14.44
In a game of pool, ball A is moving with the velocity
00
vvi
when it strikes balls B and C, which are at rest side by side.
Assuming frictionless surfaces and perfectly elastic impact (i.e.,
conservation of energy), determine the final velocity of each
ball, assuming that the path of A is (a) perfectly centered and
that A strikes B and C simultaneously, (b) not perfectly centered
and that A strikes B slightly before it strikes C.
SOLUTION
(a) A strikes B and C simultaneously:
000 0
3
AAA A
PROBLEM 14.44 (Continued)
PROBLEM 14.44 (Continued)
Second impact: A strikes C.
0
C
PROBLEM 14.45
The 2-kg sub-satellite B has an initial velocity
vB
= 3 m/s
j
. It is connected to the
20-kg base-satellite A by a 500-m space tether. Determine the velocity of the base
satellite and sub-satellite immediately after the tether becomes taut (assuming no
rebound).
SOLUTION
B
PROBLEM 14.46
A 900-lb space vehicle traveling with a velocity 0(1500 ft/s)vk
passes through the origin O. Explosive
charges then separate the vehicle into three parts A, B, and C, with masses of 150 lb, 300 lb, and 450 lb,
respectively. Knowing that shortly thereafter the positions of the three parts are, respectively, A(250, 250, 2250),
B (600, 1300, 3200), and C (–475, –950, 1900), where the coordinates are expressed in ft, that the velocity of B
is (500 ft/s) (1100 ft/s) (2100 ft/s) ,
B vijk
and that the x component of the velocity of C is 400 ft/s,
determine the velocity of part A.
SOLUTION
B
PROBLEM 14.46 (Continued)
ijk ijk ijk
Avijk
PROBLEM 14.47
Four small disks A, B, C, and D can slide freely on a frictionless horizontal
surface. Disks B, C, and D are connected by light rods and are at rest in the
position shown when disk B is struck squarely by disk A, which is moving to
the right with a velocity 0
v (38.5 ft/s)i. The weights of the disks are
A
W B
W C
W 15 lb, and D
W 30 lb. Knowing that the velocities of the
disks immediately after the impact are
A
v B
v (8.25 ft/s)i, C
v C
vi, and
D
v D
vi, determine (a) the speeds C
v and ,
D
v (b) the fraction of the
initial kinetic energy of the system which is dissipated during the collision.
SOLUTION
PROBLEM 14.48
In the scattering experiment of Problem 14.26, it is known that
the alpha particle is projected from A0(300, 0, 300) and that it
collides with the oxygen nucleus C at (240, 200, 100),Q where
all coordinates are expressed in millimeters. Determine the
coordinates of Point 0
B
where the original path of nucleus B
intersects the zx plane. (Hint: Express that the angular
momentum of the three particles about Q is conserved.)
PROBLEM 14.26 In a scattering experiment, an alpha particle A
is projected with the velocity 0(600 m/s)
ui
(750 m/s)
j
(800 m/s)k 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 1(280, 240, 120)A and A2 (360, 320,
160), while nuclei B and C are observed to move along paths
defined, respectively, by 1(147, 220, 130),B 2(114, 290,120),B
and by C1(240, 232, 90) and 2(240, 280, 75).C 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.
SOLUTION
Conservation of angular momentum about Q:
PROBLEM 14.48 (Continued)
BQ
B
PROBLEM 14.49
Three identical small spheres, each of weight 2 lb, can slide
freely on a horizontal frictionless surface. Spheres B and C
are connected by a light rod and are at rest in the position
shown when sphere B is struck squarely by sphere A which is
moving to the right with a velocity v0 (8 ft/s)i. Knowing
that θ = 45° and that the velocities of spheres A and B
immediately after the impact are A
v 0 and
B
v (6 ft/s)i
()
By
v j, determine ()
B
y
v and the velocity of C immediately
after impact.
SOLUTION
Let m be the mass of one ball.
PROBLEM 14.50
Three small spheres A, B, and C, each of mass m, are connected to a
small ring D of negligible mass by means of three inextensible,
inelastic cords of length l. The spheres can slide freely on a frictionless
horizontal surface and are rotating initially at a speed 0
v about ring D
which is at rest. Suddenly the cord CD breaks. After the other two
cords have again become taut, determine (a) the speed of ring D,
(b) the relative speed at which spheres A and B rotate about D, (c) the
fraction of the original energy of spheres A and B that is dissipated
when cords AD and BD again became taut.
GAB A
PROBLEM 14.50 (Continued)
PROBLEM 14.51
In a game of billiards, ball A is given an initial velocity 0
v along
the longitudinal axis of the table. It hits ball B and then ball C,
which are both at rest. Balls A and C are observed to hit the sides
of the table squarely at
A
and C, respectively, and ball B is
observed to hit the side obliquely at .
B
Knowing that
04 m/s,v 1.92 m/s,
A
v and 1.65 m,a determine (a) the
velocities B
v and C
v of balls B and C, (b) the Point C
where
ball C hits the side of the table. Assume frictionless surfaces and
perfectly elastic impacts (that is, conservation of energy).
SOLUTION
Velocities in m/s. Lengths in meters. Assume masses are 1.0 for each ball.
PROBLEM 14.51 (Continued)
PROBLEM 14.52
For the game of billiards of Problem 14.51, it is now assumed
that 05v
m/s, 3.2
C
v m/s, and 1.22c m. Determine
(a) the velocities
A
v and B
v of balls A and B, (b) the Point
A
where ball A hits the side of the table.
PROBLEM 14.51 In a game of billiards, ball A is given an initial
velocity 0
v along the longitudinal axis of the table. It hits ball B
and then ball C, which are both at rest. Balls A and C are
observed to hit the sides of the table squarely at
A
and C
,
respectively, and ball B is observed to hit the side obliquely at .
B
Knowing that 04 m/s,v 1.92 m/s,
A
v and 1.65 m,a
determine (a) the velocities B
v and C
v of balls B and C, (b) the
Point C
where ball C hits the side of the table. Assume
frictionless surfaces and perfectly elastic impacts (that is,
conservation of energy).
SOLUTION
Velocities in m/s. Lengths in meters. Assume masses are 1.0 for each ball.
PROBLEM 14.53
Two small disks A and B, of mass 3 kg and
1.5 kg, respectively, may slide on a
horizontal, frictionless surface. They are
connected by a cord, 600 mm, long, and
spin counterclockwise about their mass
center G at the rate of 10 rad/s. At
0,t
the
coordinates of G are 00
0, 2 m,xy
and
its velocity is 0
(1.2 m/s) (0.96 m/s) .vi j
Shortly thereafter, the cord breaks; disk A is
then observed to move along a path parallel
to the y axis and disk B along a path which
intersects the x axis at a distance b
7.5 m
from O. Determine (a) the velocities of A
and B after the cord breaks, (b) the distance
a from the y-axis to the path of A.
SOLUTION
PROBLEM 14.53 (Continued)
Kinetic energy: Using Eq. (14.29),
22222
00 0
22 2 2
11 11 1
:
22 22 2
111
iAABB
i
Tmv mv mv mv mv
PROBLEM 14.54
Two small disks A and B, of mass 2 kg and 1 kg,
respectively, may slide on a horizontal and
frictionless surface. They are connected by a cord
of negligible mass and spin about their mass center
G. At
0,t
G is moving with the velocity 0
v
and
its coordinates are 0
x
0, 0
1.89 m.y
Shortly
thereafter, the cord breaks and disk A is observed to
move with a velocity
(5 m/s)
A
vj
in a straight
line and at a distance
2.56 ma
from the y-axis,
while B moves with a velocity
(7.2 m/s) (4.6 m/s)
B
vij
along a path
intersecting the x-axis at a distance
7.48 mb
from the origin O. Determine (a) the initial velocity
0
v
of the mass center G of the two disks, (b) the
length of the cord initially connecting the two disks,
(c) the rate in rad/s at which the disks were spinning
about G.
0
222
00
22323
31
23
Tvl
PROBLEM 14.54 (Continued)
PROBLEM 14.55
Three small identical spheres A, B, and C, which can slide on a horizontal, frictionless surface, are attached to
three 9-in-long strings, which are tied to a ring G. Initially, the spheres rotate clockwise about the ring with
a relative velocity of 2.6 ft/s and the ring moves along the x-axis with a velocity
0
v
(1.3 ft/s)
i
.
Suddenly,
the ring breaks and the three spheres move freely in the xy plane with A and B, following paths parallel to the
y-axis at a distance 1.0 ft
a
from each other and C following a path parallel to the x-axis. Determine
(a) the velocity of each sphere, (b) the distance d.
SOLUTION
