2 1
12–21.
A freight train travels at where
t
is the
elapsed time in seconds. Determine the distance traveled in
three seconds, and the acceleration at this time.
v
=6011–
e
–
t
2ft>s,
SOLUTION
s
v
2 2
12–22.
SOLUTION
Asandbag is dropped from a balloon which is ascending
vertically at a constant speed of .If the bag is released
with the same upward velocity of when and hits
the ground when ,determine the speed of the bag as
it hits the ground and the altitude of the balloon at this
instant.
t=8s
t=06m>s
6m>s
2 3
12–23.
A particle is moving along a straight line such that its
acceleration is defined as a = (–2v) m
>
s2, where v is in
meters per second. If v = 20 m
s when s = 0 and t = 0,
determine the particle’s position, velocity, and acceleration
as functions of time.
SOLUTION
2 4
*12–24.
The acceleration of a particle traveling along a straight line
vwhen , determine the particle’s elocity at m.s=2t=0
i sis in meters. If mv=0, s=1s a=1
4s1>2m>,where s2
SOLUTION
2 5
12–25.
If t
h
e effects of atmosp
h
er
i
c res
i
stance are accounte
d
for,a
falling body has an acceleration defined by the equation
, where is in and the
positive direction is downward. If the body is released from
rest at a very high altitude, determine (a) the velocity when
, and (b) the body’s terminal or maximum attainable
velocity (as ).t:q
t=5s
m>sva =9.81[1 –v2(10–4)] m>s2
SOLUTION
2 6
12–26.
T
h
e acce
l
erat
i
on of a part
i
c
l
e a
l
ong a stra
i
g
h
t
li
ne
i
s
d
ef
i
ne
d
by where tis in seconds. At
and When determine (a) the
particle’s position, (b) the total distance traveled, and
(c) the velocity.
t=9s,v=10 m>s.s=1m
t=0,a=12t–92m>s2,
2 7
12–27.
When a particle falls through the air, its initial acceleration
diminishes until it is zero, and thereafter it falls at a
constant or terminal velocity . If this variation of the
acceleration can be expressed as
determine the time needed for the velocity to become
Initially the particle falls from rest.v=vf>2.
a=1g>v2f21v2f–v22,
vf
a=g
2 8
*12–28.
SOLUTION
Two particles Aand Bstart from rest at the origin and
move along a straight line such that and
, where tis in seconds. Determine the
distance between them when and the total distance
each has traveled in .t=4s
t=4s
aB=(12t2–8) ft>s2
aA=(6t–3) ft>s2
s
=0
2 9
12–29.
A ball A is thrown vertically upward from the top of a
30-m-high building with an initial velocity of 5 m
s. At the
same instant another ball B is thrown upward from
theground with an initial velocity of 20 m
>
s. Determine the
height from the ground and the time at which they pass.
SOLUTION
3 0
12–30.
Asphere is fired downwards into a medium with an initial
speed of . If it experiences a deceleration of
where tis in seconds, determine the
distance traveled before it stops.
a=(–6t)m>s2,
27 m>s
SOLUTION
3 1
12–31.
The velocity of a particle traveling along a straight line is
, where kis constant. If when ,
determine the position and acceleration of the particle as a
function of time.
t=0s=0v=v0–ks
SOLUTION
Ans:
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issue is due to PDF
compression.>
3 2
*12–32.
SOLUTION
Ball Ais thrown vertically upwards with a velocity of .
Ball Bis thrown upwards from the same point with the
same velocity tseconds later. Determine the elapsed time
from the instant ball Ais thrown to when the
balls pass each other, and find the velocity of each ball at
this instant.
t62v0>g
v0
3 3
12–33.
As a body is projected to a high altitude above the earth’s
surface, the variation of the acceleration of gravity with
respect to altitude ymust be taken into account. Neglecting
air resistance, this acceleration is determined from the
formula , where is the constant
gravitational acceleration at sea level, Ris the radius of the
earth, and the positive direction is measured upward. If
and , determine the minimum
initial velocity (escape velocity) at which a projectile should
be shot vertically from the earth’s surface so that it does not
fall back to the earth. Hint: This requires that as
y:q.
v=0
R=6356 kmg0=9.81 m>s2
g0
a=-g0[R2>(R+y)2]
SOLUTION
3 4
12–34.
Accounting for the variation of gravitational acceleration
a with respect to altitude y (see Prob. 12–36), derive an
equation that relates the velocity of a freely falling particle
to its altitude. Assume that the particle is released from
rest at an altitude y0 from the earth’s surface. With what
velocity does the particle strike the earth if it is released
from rest at an altitude y0 = 500 km? Use the numerical
data in Prob. 12–36.
3 5
12–35.
Afreight train starts from rest and travels with a constant
acceleration of .After a time it maintains a
constant speed so that when it has traveled 2000 ft.
Determine the time and draw the –tgraph for the motion.
vt¿
t=160 s
t¿0.5 ft>s2
SOLUTION
3 6
*12–36.
The s–t graph for a train has been experimentally
determined. From the data, construct the v–t and a–t graphs
for the motion; 0 … t … 40 s. For 0 … t … 30 s, the curve is
s = (0.4t2) m, and then it becomes straight for t Ú 30 s.
SOLUTION
s (m)
600
360
3 7
12–37.
Two rockets start from rest at the same elevation. Rocket A
accelerates vertically at 20 m
>
s2 for 12 s and then maintains
a constant speed. Rocket B accelerates at 15 m
s2 until
reaching a constant speed of 150 m
>
s. Construct the a–t, v–t,
and s–t graphs for each rocket until t = 20 s. What is the
distance between the rockets when t = 20 s?
SOLUTION
3 8
12–38.
A particle starts from and travels along a straight line
with a velocity , where is in
seconds.Construct the and graphs for the time
interval .0 …t…4 s
a–tv–t
tv =(t2–4t+3) m > s
s=0
SOLUTION
3 9
12–39.
SOLUTION
If the position of a particle is defined by
the ,,and graphs for .0 …t…10 sa–tv–ts–t
s=[2 sin (p>5)t+4] , where tis in seconds,constructm
4 0
*12–40.
An airplane starts from rest, travels
5
000 ft down a runway,
and after uniform acceleration, takes off with a speed of
It then climbs in a straight line with a uniform
acceleration of until it reaches a constant speed of
Draw the s–t,v–t,and a–tgraphs that describe
the motion.
220 mi>h.
3ft>s2
162 mi>h.
SOLUTION
Ans:
s=12943.34 ft
v3=v2+ac t
t=28.4 s