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Problem 13.1 In Example 13.2, suppose that the vehi-
cle is dropped from a height h=6m. (a) What is the
downward velocity 1 s after it is released? (b) What is
its downward velocity just before it reaches the ground?
Solution: The equations that govern the motion are:
a=−g=−9.81 m/s2
v=−gt
Problem 13.2 The milling machine is programmed so
that during the interval of time from t=0tot=2s,
the position of its head (in inches) is given as a function
of time by s=4t−2t3. What are the velocity (in in/s)
and acceleration (in in/s2) of the head at t=1s?
Solution: The motion is governed by the equations
Problem 13.3 In an experiment to estimate the accel-
eration due to gravity, a student drops a ball at a distance
of 1 m above the floor. His lab partner measures the time
it takes to fall and obtains an estimate of 0.46 s.
(a) What do they estimate the acceleration due to grav-
ity to be?
(b) Let sbe the ball’s position relative to the floor.
Using the value of the acceleration due to gravity
that they obtained, and assuming that the ball is
released at t=0, determine s(in m) as a function
of time.
s
Solution: The governing equations are
(b) The distance sis then given by
1
Problem 13.4 The boat’s position during the interval
of time from t=2stot=10 s is given by s=4t+
1.6t2−0.08t3m.
(a) Determine the boat’s velocity and acceleration at
t=4s.
(b) What is the boat’s maximum velocity during this
interval of time, and when does it occur?
Solution:
10
Problem 13.5 The rocket starts from rest at t=0 and
travels straight up. Its height above the ground as a
function of time can be approximated by s=bt2+ct3,
where band care constants. At t=10 s, the rocket’s
velocity and acceleration are v=229 m/s and a=28.2
m/s2. Determine the time at which the rocket reaches
supersonic speed (325 m/s). What is its altitude when
that occurs?
Solution: The governing equations are
s=bt2+ct3,
Problem 13.6 The position of a point during the inter-
val of time from t=0tot=6 s is given by s=−
1
2t3+
6t2+4tm.
(a) What is the maximum velocity during this interval
of time, and at what time does it occur?
(b) What is the acceleration when the velocity is a
maximum?
Problem 13.7 The position of a point during the inter-
val of time from t=0tot=3 seconds is s=12 +
5t2−t3ft.
(a) What is the maximum velocity during this interval
of time, and at what time does it occur?
(b) What is the acceleration when the velocity is a
maximum?
Problem 13.8 The rotating crank causes the position
t=0.375 s.
(b) What is the maximum magnitude of the velocity
of P?
(c) When the magnitude of the velocity of Pis a
maximum, what is the acceleration of P?
s
v=ds
dt =0.8πcos(2πt) ⇒
a(0.375)=−11.2 m/s2
b) vmax =0.8π=2.513 m/s2
Problem 13.9 For the mechanism in Problem 13.8,
draw graphs of the position s, velocity v, and acce-
Solution:
12
c
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Problem 13.10 A seismograph measures the horizon-
tal motion of the ground during an earthquake. An engi-
neer analyzing the data determines that for a 10-s interval
of time beginning at t=0, the position is approximated
by s=100 cos(2πt) mm. What are (a) the maximum
velocity and (b) maximum acceleration of the ground
during the 10-s interval?
ds
dt t=(2n−1)
4
=[0.2π]=0.628 m/s.
dt t=nπ
2
Problem 13.11 In an assembly operation, the robot’s
arm moves along a straight horizontal line. During an
interval of time from t=0tot=1 s, the position of the
arm is given by s=30t2−20t3mm. (a) Determine the
maximum velocity during this interval of time. (b) What
are the position and acceleration when the velocity is a
maximum?
s
Solution:
v=(60)1
Problem 13.12 In Active Example 13.1, the accelera-
tion (in m/s2) of point Prelative to point Ois given as
Solution: The governing equations are
a=(3 m/s4)t2
Problem 13.13 The Porsche starts from rest at time
t=0. During the first 10 seconds of its motion, its
velocity in km/h is given as a function of time by v=
22.8t−0.88t2, where tis in seconds. (a) What is the
car’s maximum acceleration in m/s2, and when does it
occur? (b) What distance in km does the car travel dur-
ing the 10 seconds?
Solution: First convert the numbers into meters and seconds
14
Problem 13.14 The acceleration of a point is a=
20tm/s2. When t=0,s =40 m and v=−10 m/s.
What are the position and velocity at t=3s?
where C2is the constant of integration.
s=(10t2−10)dt +C2=10
3t3−10t+C2.
Problem 13.15 The acceleration of a point is a=
60t−36t2ft/s2. When t=0,s =0 and v=20 ft/s.
What are position and velocity as a function of time?
Problem 13.16 As a first approximation, a bioengineer
studying the mechanics of bird flight assumes that the
snow petrel takes off with constant acceleration. Video
measurements indicate that a bird requires a distance of
4.3 m to take off and is moving at 6.1 m/s when it does.
What is its acceleration?
Solution: The governing equations are
Problem 13.17 Progressively developing a more real-
istic model, the bioengineer next models the acceleration
of the snow petrel by an equation of the form a=
C(1+sin ωt), where Cand ωare constants. From video
measurements of a bird taking off, he estimates that
ω=18/s and determines that the bird requires 1.42 s
to take off and is moving at 6.1 m/s when it does. What
is the constant C?
16
Problem 13.18 Missiles designed for defense against
ballistic missiles have attained accelerations in excess
of 100 g’s, or 100 times the acceleration due to gravity.
Suppose that the missile shown lifts off from the ground
and has a constant acceleration of 100 g’s. How long
does it take to reach an altitude of 3000 m? How fast is
it going when it ranches that altitude?
Problem 13.19 Suppose that the missile shown lifts
off from the ground and, because it becomes lighter as
its fuel is expended, its acceleration (in g’s) is given as
a function of time in seconds by
a=100
1−0.2t.
What is the missile’s velocity in miles per hour 1 s after
liftoff?
Problem 13.20 The airplane releases its drag para-
chute at time t=0. Its velocity is given as a function
of time by
v=80
1+0.32tm/s.
What is the airplane’s acceleration at t=3s?
Problem 13.21 How far does the airplane in Problem
Solution:
Problem 13.22 The velocity of a bobsled is v=
10tft/s. When t=2 s, the position is s=25 ft. What
is its position at t=10 s?
Solution: The equation for straight line displacement under con-
stant acceleration is
18
Problem 13.23 In September, 2003, Tony Schumacher
Solution:
Problem 13.24 The velocity of an object is v=200 −
Solution: The acceleration is
3
3t36
3
Problem 13.25 An inertial navigation system mea-
Solution:
Problem 13.26 In Example 13.3, suppose that the chee-
tah’s acceleration is constant and it reaches its top speed
of 75 mi/h in 5 s. What distance can it cover in 10 s?
Problem 13.27 The graph shows the airplane’s accel-
eration during its takeoff. What is the airplane’s velocity
when it rotates (lifts off) at t=30 s?
a
9 ft/s2
3 ft/s2
5 s0
30 s
Solution: Velocity =Area under the curve
v=1
Problem 13.28 Determine the distance traveled during
its takeoff by the airplane in Problem 13.27.
Solution: for 0 ≤t≤5s
a=6 ft/s2
20
Problem 13.29 The car is traveling at 30 mi/h when
the traffic light 295 ft ahead turns yellow. The driver
takes one second to react before he applies the brakes.
(a) After he applies the brakes, what constant rate of
deceleration will cause the car to come to a stop
just as it reaches the light?
(b) How long does it take the car to travel the 295 ft?
295 ft
30 mi/h
Problem 13.30 The car is traveling at 30 mi/h when
the traffic light 295 ft ahead turns yellow. The driver
takes 1 s to react before he applies the accelerator. If
the car has a constant acceleration of 5 ft/s2and the
light remains yellow for 5 s, will the car reach the light
before it turns red? How fast is the car moving when it
reaches the light?
Problem 13.31 A high-speed rail transportation sys-
tem has a top speed of 100 m/s. For the comfort of the
passengers, the magnitude of the acceleration and decel-
eration is limited to 2 m/s2. Determine the time required
for a trip of 100 km.
A plot of velocity versus time can be made and the area under the
Problem 13.32 The nearest star, Proxima Centauri, is
4.22 light years from the Earth. Ignoring relative motion
between the solar system and Proxima Centauri, sup-
pose that a spacecraft accelerates from the vicinity of
the Earth at 0.01 g (0.01 times the acceleration due to
gravity at sea level) until it reaches one-tenth the speed
of light, coasts until it is time to decelerate, then decel-
erates at 0.01 g until it comes to rest in the vicinity of
Proxima Centauri. How long does the trip take? (Light
travels at 3 ×108m/s.)
reach 0.1 times the speed of light is
The total time of flight is ttotal =t1+t2+t3=1.63751 ×109sec-
22
Problem 13.33 A race car starts from rest and accel-
erates at a=5+2tft/s2for 10 seconds. The brakes
are then applied, and the car has a constant acceler-
ation a=−30 ft/s2until it comes to rest. Determine
(a) the maximum velocity, (b) the total distance trav-
eled; (c) the total time of travel.
from which
vmax =150 ft/s .
(b) The distance traveled in the first interval is
s(10)=10
0(5t+t2)dt =5
2t2+1
3t310
0
=583.33 ft.
The time of travel in the second interval is
v(t2−10)=0=a(t2−10)+v(10), t2≥10 s,
from which
(t2−10)=−150
−30 =5,and
Problem 13.34 When t=0, the position of a point is
s=6 m and its velocity is v=2 m/s. From t=0tot=
6 s, the acceleration of the point is a=2+2t2m/s2.
From t=6 s until it comes to rest, its acceleration is
a=−4 m/s2.
(a) What is the total time of travel?
(b) What total distance does the point move?
Problem 13.35 Zoologists studying the ecology of the
run 65 km/h. If the animals run along the same straight
Solution: The top speeds are Vc=100 km/h =27.78 m/s for the
4=6.94 m/s2for the cheetah, and as=Vs
4=4.513 m/s2for the
springbuck. Divide the intervals into the acceleration phase and the
Problem 13.36 Suppose that a person unwisely drives
75 mi/h in a 55 mi/h zone and passes a police car going
55 mi/h in the same direction. If the police officers begin
constant acceleration at the instant they are passed and
increase their speed to 80 mi/h in 4 s, how long does it
take them to be even with the pursued car?
Solution: The conversion from mi/h to ft/s is
The distance traveled by the pursued car during this acceleration is
Problem 13.37 If θ=1 rad and dθ
dt =1 rad/s, what
is the velocity of Prelative to O?
24
Problem 13.38 In Problem 13.37, if θ=1 rad, dθ/dt
=−2 rad/s and d2θ/dt2=0, what are the velocity and
Solution: The velocity is
Problem 13.39 If θ=1 rad and dθ
dt =1 rad/s, what
is the velocity of Prelative to O?
O
P
200 mm 400 mm
Solution: The acute angle formed by the 400 mm arm with the
horizontal is given by the sine law:
From the expression for the angle
Problem 13.40 In Active Example 13.4, determine the
time required for the plane’s velocity to decrease from
50 m/s to 10 m/s.
Solution: From Active Example 13.4 we know that the accelera-
Problem 13.41 An engineer designing a system to con-
trol a router for a machining process models the sys-
tem so that the router’s acceleration (in in/s2) during an
interval of time is given by a=−0.4v, where vis the
velocity of the router in in/s. When t=0, the position is
s=0 and the velocity is v=2 in/s. What is the position
at t=3s?
Solution: We will first find the velocity at t =3.
26
Problem 13.42 The boat is moving at 10 m/s when
its engine is shut down. Due to hydrodynamic drag, its
subsequent acceleration is a=−0.05v2m/s2, where v
is the velocity of the boat in m/s. What is the boat’s
velocity 4 s after the engine is shut down?
Solution:
Problem 13.43 In Problem 13.42, what distance does
Solution: From Problem 13.42 we know
Problem 13.44 A steel ball is released from rest in a
container of oil. Its downward acceleration is a=2.4−
0.6vin/s2, where vis the ball’s velocity in in/s. What is
the ball’s downward velocity 2 s after it is released?
Solution:
Problem 13.45 In Problem 13.44, what distance does
Solution: From 13.44 we know
Problem 13.46 The greatest ocean depth yet discov-
ered is the Marianas Trench in the western Pacific Ocean.
A steel ball released at the surface requires 64 minutes
to reach the bottom. The ball’s downward acceleration
is a=0.9g−cv, where g=9.81 m/s2and the constant
c=3.02 s−1. What is the depth of the Marianas Trench
in kilometers?
Solution:
Integrating,
Problem 13.47 The acceleration of a regional airliner
Solution:
Problem 13.48 In Problem 13.47, what distance does
the airliner require to take off?
Solution:
28
