+

1 2.3
+
-12-49. A particle travels along a curve defined by the
equation s = (t3 - 3 t2 2t) m. where t is in seconds. Draw
the s - t, v - t, and a - t graphs for the particle for
o ::; t ::; 3 s.
27
RECTILINEAR KINEMATICS: ERRATIC MOTION
*12-52. A car travels up a hill with the speed shown.
Determine the total distance the car travels until it stops
( t = 60 s ) . Plot the a-t graph.
v (rnls)
12-50. A truck is traveling along the straight line with a
velocity described by the graph. Construct the a-s graph
for 0 ::; s ::; 1 500 ft.
10
v (fIls)
v=
'-------+--"!--
s3/4
30
Prob. 12-52
0.
6
�-��---�
75
I
1 (s)
v (rnls)
1500
Prob. 12-50
12-51. A car starts from rest and travels along a straight
road with a velocity described by the graph. Determine the
total distance traveled until the car stops. Construct the s-t
and a-t graphs.
v(rnls)
60
-12-53. The snowmobile moves along a straight course
according to the v-t graph. Construct the s-t and a-t graphs
for the same 50-s time interval. When t = 0, s = O.
<-------r- s(fl)
625
+---------.....
"�:ias
30 60 90 "
Prob. 12-51
+
30
Prob. 12-53
50
(s)
12-54. A motorcyclist at A is traveling at 60 ft/s when he
wishes to pass the truck T which is traveling at a constant
speed of 60 ft/s. To do so the motorcyclist accelerates at
6 ft/S2 until reaching a maximum speed of 85 ft/s. If he then
maintains this speed, determine the time needed for him to
reach a point located 100 ft in front of the truck. Draw the
v-t and s-t graphs for the motorcycle during this time.
.,
45
IL...__
__
__
__
__
--'__
__
__
----'
__
__
_ I
(VIII) ]
=
60
fIls
I(S )
Prob. 12-54
•
28
K I N E M AT I C S O F A PAR T I C L E
CHAPTER 1 2
12-55. An airplane traveling at 70 mls lands on a straight
runway and has a deceleration described by the graph.
Determine the time t' and the distance traveled for it to
reach a speed of 5 m/s. Construct the v-t and s-t graphs for
this time interval, 0 :0; t :0; t'.
o(m/s2)
o(m/s2)
�
t'
5
I
-4
-10
-12-57. The dragster starts from rest and travels along a
straight track with an acceleration-deceleration described
by the graph. Construct the v-s graph for 0 :0; S :0; s', and
determine the distance s' traveled before the dragster again
comes to rest.
I(S)
a = o.IS
;
25
-
- 15
�
V1
s'
200
s Cm )
Prob. 12-55
Prob. 12-57
*12-56. The position of a cyclist traveling along a straight
road is described by the graph. Construct the v-t and a-t
graphs.
12-58. A sports car travels along a straight road with an
acceleration-deceleration described by the graph. If the car
starts from rest, determine the distance s' the car travels
until it stops. Construct the v-s graph for 0 :0; S :0; s'.
S (m)
1 37.5
S
= - 0.625 ?
+
27.51 - 1 62.5
6 -j-------,
--+---,S--1f-----+-,-- (ft)
1000
50
'-"'---+----+- I (s)
10
-4
20
Prob. 12-56
Prob. 12-58
S'
1 2.3
12-59. A missile starting from rest travels along a straight
track and for 10 s has an acceleration as shown. Draw the
v-t graph that describes the motion and find the distance
traveled in 10 s.
+-------.,
40
30 --'----�
29
RECTILINEAR KINEMATICS: ERRATIC MOTION
-12-61. The v-t graph of a car while traveling along a road
is shown. Draw the s-t and a-t graphs for the motion.
v (m/s)
a=
a
=
21 + 20
20
61
'----+--+-------- /(S )
+---r-----�
"---- +------+-------'-- / (s)
5
20
30
5 10
Prob. 12-59
Prob. 12-61
*12-60. A motorcyclist starting from rest travels along a
straight road and for 10 s has an acceleration as shown.
Draw the v-t graph that describes the motion and find the
distance traveled in 10 s.
12-62. The boat travels in a straight line with the
acceleration described by the a-s graph. If it starts from rest,
construct the v-s graph and determine the boat's maximum
speed. What distance S' does it travel before it stops?
6
+------�--�
'"-"""""----+- (s)
6
Prob. 12-60
10 1
6 �.02S+ 6
3
150
-4
Prob. 12-62
S'
sCm)
•
•
30
CHAPTER 1 2
K I N E M AT I C S O F A P A RT I C L E
12-63. The rocket has a n acceleration described by the
graph. If it starts from rest, construct the v-( and s-(
graphs for the motion for the time interval 0 :S ( :S 1 4 s .
38
-12-65. The acceleration of the speed boat starting from
rest is described by the graph. Construct the v-s graph.
d��
10
18
'-------i----i---- t(s)
14
9
i
200
s(ft)
Prob. 12-65
Prob. 12-63
*12-64. The jet bike is moving along a straight road with the
speed described by the v- s graph. Construct the a-s graph.
12-66. The boat travels along a straight line with the speed
described by the graph. Construct the s-( and a-s graphs.
Also, determine the time required for the boat to travel a
distance s = 400 m if s = 0 when ( = o.
v(rnls)
v(m/s)
75 -
i
500
v
= 5s 1/2
80
v = -0.2s
15
225
v = 0.2s ______
+ 120
525
s Cm)
L----,--- s ( m)
100
Prob. 12-64
400
Prob. 12-66
1 2.3
12-67. The s-t graph for a train has been determined
experimentally. From the data, construct the v-t and a-t
graphs for the motion.
s (m)
31
RECTILINEAR KINEMATICS: ERRATIC MOTION
-12-69. The airplane travels along a straight runway with
an acceleration described by the graph. If it starts from rest
and requires a velocity of 90 m/s to take off, determine the
minimum length of runway required and the time t ' for take
off. Construct the v-t and s-t graphs.
10 0 0 0 0 0 0 0 o� lo 0 0 0 0 0 0 0 0�1
600 1-------,
8 4-----�-----,
360 1-------/
a = 0.8t
"------f--;--
"-=-------'------'-- ---- t (s)
10
Prob. 12-69
Prob. 12-67
*12-68. The airplane lands at 250 ft/s on a straight runway
and has a deceleration described by the graph. Determine
the distance s' traveled before its speed is decreased to
25 ft/ s. Draw the s-t graph.
�op?
a(ft/s2)
1750
-15
t(s)
40
30
-7.5
t'
I
I
Prob. 12-68
12-70. The a-t graph of the bullet train is shown. If the
train starts from rest, determine the elapsed time t ' before it
again comes to rest. What is the total distance traveled
during this time interval? Construct the v-t and s-t graphs.
a(m/s2)
s'
c::=:±:E
!
�
s (ft)
I
a = - ( I5)t
+5
3
t'
30
75
Prob. 12-70
t(s)
•
A N S W E R S TO S E L E C T E D P R O B L E M S
700
12-38.
12-39.
12-41.
12-42.
12-45.
12-46.
12-49.
12-51.
12-53.
12-54.
12-55.
12-57.
12-58.
12-59.
12-61.
12-62.
12-63.
12-65.
12-66.
12-69.
12-70.
12-71.
VA = � gt t
VB = � gt i
V = 11.2 km/s
V = 3.02 km/s t
V = -30t + 15t2 m/s
At rest at t = 0 and t = 2 s
Stot = 30 m
vavg = 15 m/s
ST = 980 m
V = 27T5 cos 1L5 t
• 1T
a = - 21T2
25 sm 5 t
Vmax = 16.7 m/s
V = 3t2 - 6t + 2
a = 6t - 6
S I { �90 s = 1350 m
S = (� t2 ) m and S = (12t - 180) m
a = 0.4 m/s2 and a = 0
t = 9.88 s
t'
I
=
8.75 S
272 m
v = (VO.1s2 + lOs) m/s and
v = (V-30s + 12 000) m/s
S' = 400 m
S' = 2500 ft
s = 917 m
s = 2t2
S = 20t - 50
s = -t2 + 60t - 450
Vmax = 36.7 m/s
S' = 319 m
v = 4t3/2 and v = 2t2 - 18t + 108
s = �t5/2 and s = �t3 - 9t2 + 108t - 340
v = VO.04s 2 + 4s ft/s
v = V20s - 1600 ft/s
t = 16.9 s
v = 0.8t, v = 24.0
a = 0.8, a = 0
v = (0.4t2 ) m/s
v = (8t - 40) m/s
t' = 16.25 s
S I { �16.25 = 540 m
t' = 133 s, s = 8857 m
v = 36.1 m/s
a = 36.5 m/s2
S { �8.75 s =
s
H.t3/
3b
3 H. 1
12-73. x =
2
ax =
4" 3b Vi
ay = 2et
12-74. a = 80.2 m/s2
(42.7, 16.0, 14.0) m
±4r cos2t
ay = -4r sin 2t
v = { -10 sin 2ti + 8 cos 2tj} m/s
a = { -20 cos 2ti - 16 sin 2tj} m/s2
V = 9.68 m/s
a = 16.8 m/s2
v = 10.4 m/s
a = 38.5 m/s2
Vx = 3.58 mis, Vy = 1.79 m/s
ax = 0.32 m/s2
ay = 0.64 m/s2 t
rB = {21.2li - 21.21j} m
rc = {28.98i - 7.765j} m
(VBc)avg = {3.88i + 6.72j } m/s
12-75. ax =
12-77.
12-78.
12-79.
12-81.
12-82. S
=
9 km
6.71 km
vavg = 4.86 m/s
(vsp)avg = 6.52 m/s
12-83. v = Ve2 k2 + b2
a = ek2
12-85. Vy = Vx - 2�0 Vx
v = 2.69 ft/s
1
( V2x + xax )
ay - a x - 200
a = 0.0200 ft/s2
12-86. Vx = va [1 + (f e ) 2 cos2 (f x) ]-t
v
Vy = o:e ( cos fx )[ 1 + ( f e ) 2 cos2 ( f x ) ] -t
D.r
=
_
= 6.49 m/s
t = 0.890 s
VA cos {} = 20
VA sin {} = 23.3
{} = 49.4°
VA = 30.7 ft/s
VB = 76.0 ft/s
{} = 57.6°
x = 222 m
y = 116 m
s = 8.68 ft
s = 34.4 ft
12-87. VA
12-89.
12-91.
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PROBLEMS
v0 = 45 km/h
11.33 A motorist enters a freeway at 45 km/h and accelerates uniformly
to 99 km/h. From the odometer in the car, the motorist knows
that she traveled 0.2 km while accelerating. Determine (a) the
acceleration of the car, (b) the time required to reach 99 km/h.
11.34 A truck travels 220 m in 10 s while being decelerated at a constant
Fig. P11.33
rate of 0.6 m/s2. Determine (a) its initial velocity, (b) its final velocity, (c) the distance traveled during the first 1.5 s.
v0
a = 0.6 m/s2
Fig. P11.34
11.35 Assuming a uniform acceleration of 11 ft/s2 and knowing that the
speed of a car as it passes A is 30 mi/h, determine (a) the time
required for the car to reach B, (b) the speed of the car as it
passes B.
vA = 30 mi/h
v1
B
A
160 ft
89.6 ft
Fig. P11.35
11.36 A group of students launches a model rocket in the vertical direc-
Fig. P11.36
v
tion. Based on tracking data, they determine that the altitude of
the rocket was 89.6 ft at the end of the powered portion of the
flight and that the rocket landed 16 s later. Knowing that the
descent parachute failed to deploy so that the rocket fell freely
to the ground after reaching its maximum altitude and assuming
that g 5 32.2 ft/s2, determine (a) the speed v1 of the rocket at
the end of powered flight, (b) the maximum altitude reached by
the rocket.
11.37 A sprinter in a 100-m race accelerates uniformly for the first 35 m
Fig. P11.37
624
and then runs with constant velocity. If the sprinter’s time for the
first 35 m is 5.4 s, determine (a) his acceleration, (b) his final
velocity, (c) his time for the race.
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Problems
11.38 A small package is released from rest at A and moves along the
skate wheel conveyor ABCD. The package has a uniform acceleration of 4.8 m/s2 as it moves down sections AB and CD, and its
velocity is constant between B and C. If the velocity of the package
at D is 7.2 m/s, determine (a) the distance d between C and D,
(b) the time required for the package to reach D.
A
625
3m
B
3m
C
d
D
Fig. P11.38
11.39 A police officer in a patrol car parked in a 70 km/h speed zone
observes a passing automobile traveling at a slow, constant speed.
Believing that the driver of the automobile might be intoxicated,
the officer starts his car, accelerates uniformly to 90 km/h in 8 s,
and, maintaining a constant velocity of 90 km/h, overtakes the
motorist 42 s after the automobile passed him. Knowing that 18 s
elapsed before the officer began pursuing the motorist, determine
(a) the distance the officer traveled before overtaking the motorist,
(b) the motorist’s speed.
11.40 As relay runner A enters the 20-m-long exchange zone with a
speed of 12.9 m/s, he begins to slow down. He hands the baton to
runner B 1.82 s later as they leave the exchange zone with the
same velocity. Determine (a) the uniform acceleration of each of
the runners, (b) when runner B should begin to run.
(vA)0 = 12.9 m/s
(vB)0 = 0
A
B
20 m
Fig. P11.40
(vA)0 = 24 mi/h
11.41 Automobiles A and B are traveling in adjacent highway lanes and
at t 5 0 have the positions and speeds shown. Knowing that
automobile A has a constant acceleration of 1.8 ft/s2 and that B
has a constant deceleration of 1.2 ft/s2, determine (a) when and
where A will overtake B, (b) the speed of each automobile at
that time.
(vB)0 = 36 mi/h
A
B
75 ft
x
Fig. P11.41
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Kinematics of Particles
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11.42 In a boat race, boat A is leading boat B by 120 ft and both boats
are traveling at a constant speed of 105 mi/h. At t 5 0, the boats
accelerate at constant rates. Knowing that when B passes A, t 5 8 s
and vA 5 135 mi/h, determine (a) the acceleration of A, (b) the
acceleration of B.
vA
120 ft
vB
A
B
Fig. P11.42
11.43 Boxes are placed on a chute at uniform intervals of time tR and
slide down the chute with uniform acceleration. Knowing that as
any box B is released, the preceding box A has already slid 6 m
and that 1 s later they are 10 m apart, determine (a) the value of
tR, (b) the acceleration of the boxes.
B
6m
(vB)0 = 0
A
vA
Fig. P11.43
11.44 Two automobiles A and B are approaching each other in adjacent
highway lanes. At t 5 0, A and B are 1 km apart, their speeds are
vA 5 108 km/h and vB 5 63 km/h, and they are at points P and
Q, respectively. Knowing that A passes point Q 40 s after B was
there and that B passes point P 42 s after A was there, determine
(a) the uniform accelerations of A and B, (b) when the vehicles
pass each other, (c) the speed of B at that time.
vA = 108 km/h
vB = 63 km/h
A
B
P
Fig. P11.44
1 km
Q
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Problems
11.45 Car A is parked along the northbound lane of a highway, and car B
is traveling in the southbound lane at a constant speed of 60 mi/h.
At t 5 0, A starts and accelerates at a constant rate aA, while at
t 5 5 s, B begins to slow down with a constant deceleration of
magnitude aA /6. Knowing that when the cars pass each other x 5
294 ft and vA 5 vB, determine (a) the acceleration aA, (b) when
the vehicles pass each other, (c) the distance d between the vehicles
at t 5 0.
A
(vB)0 = 60 mi /h
(vA)0 = 0
B
x
d
Fig. P11.45
11.46 Two blocks A and B are placed on an incline as shown. At t 5 0,
A is projected up the incline with an initial velocity of 27 ft/s and
B is released from rest. The blocks pass each other 1 s later, and
B reaches the bottom of the incline when t 5 3.4 s. Knowing that
the maximum distance from the bottom of the incline reached by
block A is 21 ft and that the accelerations of A and B (due to
gravity and friction) are constant and are directed down the
incline, determine (a) the accelerations of A and B, (b) the distance d, (c) the speed of A when the blocks pass each other.
(vB)0 = 0
B
(vA)0 = 27 ft /s
A
d
Fig. P11.46
A
C
D
11.47 Slider block A moves to the left with a constant velocity of 6 m/s.
Determine (a) the velocity of block B, (b) the velocity of portion
D of the cable, (c) the relative velocity of portion C of the cable
with respect to portion D.
B
11.48 Block B starts from rest and moves downward with a constant
acceleration. Knowing that after slider block A has moved
400 mm its velocity is 4 m/s, determine (a) the accelerations
of A and B, (b) the velocity and the change in position of B
after 2 s.
Fig. P11.47 and P11.48
627
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11.49 The elevator shown in the figure moves downward with a constant
Kinematics of Particles
velocity of 15 ft/s. Determine (a) the velocity of the cable C, (b) the
velocity of the counterweight W, (c) the relative velocity of the cable
C with respect to the elevator, (d) the relative velocity of the counterweight W with respect to the elevator.
W
C
C
E
M
A
Fig. P11.49 and P11.50
11.50 The elevator shown starts from rest and moves upward with a con-
B
stant acceleration. If the counterweight W moves through 30 ft in
5 s, determine (a) the acceleration of the elevator and the cable
C, (b) the velocity of the elevator after 5 s.
11.51 Collar A starts from rest and moves upward with a constant accelera-
tion. Knowing that after 8 s the relative velocity of collar B with
respect to collar A is 24 in./s, determine (a) the accelerations of A
and B, (b) the velocity and the change in position of B after 6 s.
11.52 In the position shown, collar B moves downward with a velocity of
Fig. P11.51
and P11.52
12 in./s. Determine (a) the velocity of collar A, (b) the velocity of
portion C of the cable, (c) the relative velocity of portion C of the
cable with respect to collar B.
11.53 Slider block B moves to the right with a constant velocity of
300 mm/s. Determine (a) the velocity of slider block A, (b) the
velocity of portion C of the cable, (c) the velocity of portion D
of the cable, (d) the relative velocity of portion C of the cable with
respect to slider block A.
C
A
B
D
Fig. P11.53 and P11.54
11.54 At the instant shown, slider block B is moving with a constant
acceleration, and its speed is 150 mm/s. Knowing that after slider
block A has moved 240 mm to the right its velocity is 60 mm/s,
determine (a) the accelerations of A and B, (b) the acceleration of
portion D of the cable, (c) the velocity and the change in position
of slider block B after 4 s.
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Problems
11.55 Block B moves downward with a constant velocity of 20 mm/s. At
t 5 0, block A is moving upward with a constant acceleration, and
its velocity is 30 mm/s. Knowing that at t 5 3 s slider block C has
moved 57 mm to the right, determine (a) the velocity of slider
block C at t 5 0, (b) the accelerations of A and C, (c) the change
in position of block A after 5 s.
C
A
B
Fig. P11.55 and P11.56
11.56 Block B starts from rest, block A moves with a constant accelera-
tion, and slider block C moves to the right with a constant acceleration of 75 mm/s2. Knowing that at t 5 2 s the velocities of B and
C are 480 mm/s downward and 280 mm/s to the right, respectively,
determine (a) the accelerations of A and B, (b) the initial velocities
of A and C, (c) the change in position of slider block C after 3 s.
B
A
11.57 Collar A starts from rest at t 5 0 and moves downward with a
constant acceleration of 7 in./s2. Collar B moves upward with a
constant acceleration, and its initial velocity is 8 in./s. Knowing that
collar B moves through 20 in. between t 5 0 and t 5 2 s, determine (a) the accelerations of collar B and block C, (b) the time at
which the velocity of block C is zero, (c) the distance through
which block C will have moved at that time.
C
11.58 Collars A and B start from rest, and collar A moves upward with an
acceleration of 3t2 in./s2. Knowing that collar B moves downward
with a constant acceleration and that its velocity is 8 in./s after moving 32 in., determine (a) the acceleration of block C, (b) the distance
through which block C will have moved after 3 s.
Fig. P11.57 and P11.58
11.59 The system shown starts from rest, and each component moves with
a constant acceleration. If the relative acceleration of block C with
respect to collar B is 60 mm/s2 upward and the relative acceleration
of block D with respect to block A is 110 mm/s2 downward, determine (a) the velocity of block C after 3 s, (b) the change in position
of block D after 5 s.
A
*11.60 The system shown starts from rest, and the length of the upper
cord is adjusted so that A, B, and C are initially at the same level.
Each component moves with a constant acceleration, and after 2 s
the relative change in position of block C with respect to block A
is 280 mm upward. Knowing that when the relative velocity of
collar B with respect to block A is 80 mm/s downward, the displacements of A and B are 160 mm downward and 320 mm downward, respectively, determine (a) the accelerations of A and B if
aB . 10 mm/s2, (b) the change in position of block D when the
velocity of block C is 600 mm/s upward.
B
C
D
Fig. P11.59 and P11.60
629