PROBLEM 11.77
An accelerometer record for the motion of a given part of a
mechanism is approximated by an arc of a parabola for 0.2 s and a
straight line for the next 0.2 s as shown in the figure. Knowing that
0v
when 0t
and x 0.8 ft when t 0.4 s, (a) construct the vt
curve for 0
t
0.4 s, (b) determine the position of the part at t
0.3 s and t 0.2 s.
SOLUTION
PROBLEM 11.78
A car is traveling at a constant speed of
54 km/h when its driver sees a child run into
the road. The driver applies her brakes until
the child returns to the sidewalk and then
accelerates to resume her original speed of
54 km/h; the acceleration record of the car is
shown in the figure. Assuming
0x
when
t
0,
determine (a) the time t
1
at which the
velocity is again 54 km/h, (b) the position of
the car at that time, (c) the average velocity of
the car during the interval 1
1
s.tt
SOLUTION
1
(b) Using the above values of the velocities, the
vt
curve is drawn as shown.
PROBLEM 11.78 (Continued)
PROBLEM 11.79
An airport shuttle train travels between two terminals that are 1.6 mi apart. To maintain passenger comfort,
the acceleration of the train is limited to 4 ft/s
2
, and the jerk, or rate of change of acceleration, is limited to
0.8 ft/s
2
per second. If the shuttle has a maximum speed of 20 mi/h, determine (a) the shortest time for the
shuttle to travel between the two terminals, (b) the corresponding average velocity of the shuttle.
SOLUTION
2
PROBLEM 11.79 (Continued)
ave
PROBLEM 11.80
During a manufacturing process, a conveyor belt starts from rest and travels a total of 1.2 ft before
temporarily coming to rest. Knowing that the jerk, or rate of change of acceleration, is limited to 4.8 ft/s
2
per
second, determine (a) the shortest time required for the belt to move 1.2 ft, (b) the maximum and average
values of the velocity of the belt during that time.
PROBLEM 11.80 (Continued)
max 4.8 ft/s /s
a
PROBLEM 11.81
Two seconds are required to bring the piston
rod of an air cylinder to rest; the acceleration
record of the piston rod during the 2 s is as
shown. Determine by approximate means (a)
the initial velocity of the piston rod, (b) the
distance traveled by the piston rod as it is
brought to rest.
SOLUTION
1. The
at curve is first approximated with a series of rectangles, each of width 0.25 s.t
The area
(t)(aave) of each rectangle is approximately equal to the change in velocity v
for the specified
where the values of ave
a and v are given in columns 1 and 2, respectively, of the following table.
2
where
12
v is the change in velocity between times t1 and t2, the velocity at the end of each
0.25 interval can be computed; see column 3 of the table and the vt
curve.
3. The
vt curve is then approximated with a series of rectangles, each of width 0.25 s. The area
ave
()( )tv of each rectangle is approximately equal to the change in position
x
for the specified
x
x
ave
PROBLEM 11.81 (Continued)
4. With
0
0x
and noting that
xx x
PROBLEM 11.82
The acceleration record shown was obtained
during the speed trials of a sports car.
Knowing that the car starts from rest,
determine by approximate means (a) the
velocity of the car at t 8 s, (b) the distance
the car has traveled at t 20 s.
SOLUTION
PROBLEM 11.82 (Continued)
PROBLEM 11.83
A training airplane has a velocity of 126 ft/s when
it lands on an aircraft carrier. As the arresting gear
of the carrier brings the airplane to rest, the
velocity and the acceleration of the airplane are
recorded; the results are shown (solid curve) in the
figure. Determine by approximate means (a) the
time required for the airplane to come to rest,
(b) the distance traveled in that time.
SOLUTION
Given:
av
curve:
0
PROBLEM 11.84
Shown in the figure is a portion of the experimentally
determined
vx
curve for a shuttle cart. Determine by
approximate means the acceleration of the cart (a) when
10 in.,x
(b) when
80 in./s.v
or
163.0 in./sa
(b) When 80 in./s,v we have
40 in./s
or
114.3 in./sa
PROBLEM 11.85
An elevator starts from rest and rises 40 m to its maximum
velocity in T s with the acceleration record shown in the figure.
Determine (a) the required time T, (b) the maximum velocity,
(c) the velocity and position of the elevator at t = T/2.
SOLUTION
Find the position of the elevator from the curve using the moment-area method.
(c) To find the position at t=T/2, first find the areas on the
at
12
and AA curveat
1
T
PROBLEM 11.85 (Continued)
0134
PROBLEM 11.86
Two road rally checkpoints A and B are located
on the same highway and are 8 mi apart. The
speed limits for the first 5 mi and the last 3 mi
are 60 mi/h and 35 mi/h, respectively. Drivers
must stop at each checkpoint, and the specified
time between points A and B is 10 min 20 s.
Knowing that a driver accelerates and
decelerates at the same constant rate, determine
the magnitude of her acceleration if she travels
at the speed limit as much as possible.
SOLUTION
2
3600
PROBLEM 11.87
As shown in the figure, from
0t
to 4 s,t the acceleration of
a given particle is represented by a parabola. Knowing that x 0
and
8 m/sv
when 0,t
(a) construct the
vt
and x–t curves
for 0
t
4 s, (b) determine the position of the particle at t 3 s.
(Hint: Use table inside the front cover.)
33.75 mx
PROBLEM 11.88
A particle moves in a straight line with the acceleration
shown in the figure. Knowing that the particle starts from
the origin with 02v
m/s, (a) construct the vt and
x
t
curves for 018t
s, (b) determine the position
and the velocity of the particle and the total distance
traveled when 18t
s.
PROBLEM 11.89
A ball is thrown so that the motion is defined by
the equations 5
x
t
and 2
26 4.9,ytt where
x and y are expressed in meters and t is expressed
in seconds. Determine (a) the velocity at t1s,
(b) the horizontal distance the ball travels before
hitting the ground.
SOLUTION
PROBLEM 11.90
The motion of a vibrating particle is defined by the position vector
32
10(1 ) (4 sin15 ) ,
tt
eet
ri
j
where
r
and t are expressed in
millimeters and seconds, respectively. Determine the velocity and
acceleration when (a)
0,t
(b)
0.5 s.t