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PROBLEM 11.19
Based on experimental observations, the acceleration of a particle is defined by the relation (0.1a sin x/b),
where a and x are expressed in m/s2 and meters, respectively. Knowing that 0.8 mb and that 1 m/sv
when 0,x determine (a) the velocity of the particle when 1 m,x
(b) the position where the velocity is
maximum, (c) the maximum velocity.
SOLUTION
dv x
PROBLEM 11.20
A spring AB is attached to a support at A and to a collar. The
unstretched length of the spring is l. Knowing that the collar is
released from rest at 0
x
x
and has an acceleration defined by
the relation 22
100( / ) axlxlx
, determine the velocity of
the collar as it passes through Point C.
0
f
PROBLEM 11.21
The acceleration of a particle is defined by the relation
1,
x
ak e
where k is a constant. Knowing that
the velocity of the particle is 9v m/s when 3x
m and that the particle comes to rest at the origin,
determine (a) the value of k, (b) the velocity of the particle when 2x
m.
SOLUTION
x
PROBLEM 11.22
Starting from
x
0 with no initial velocity, a particle is given an acceleration
2
0.1 16,av
where
a
and
v
are expressed in ft/s
2
and ft/s, respectively. Determine (
a
) the position of the particle when
v
3ft/s, (
b
) the speed and acceleration of the particle when
x
4 ft.
SOLUTION
vdv
PROBLEM 11.23
A ball is dropped from a boat so that it strikes the surface of a
lake with a speed of 16.5 ft/s. While in the water the ball
experiences an acceleration of 10 0.8 ,av
where
a
and
v
are expressed in ft/s
2
and ft/s, respectively
. Knowing the
ball takes 3 s to reach the bottom of the lake, determine (a) the
depth of the lake, (b) the speed of the ball when it hits the
bottom of the lake.
SOLUTION
10 0.8
dv
av
dt
PROBLEM 11.23 (Continued)
t
tt
PROBLEM 11.24
The acceleration of a particle is defined by the relation ,akv where k is a constant. Knowing that x 0
and v 81 m/s at t 0 and that v 36 m/s when x 18 m, determine (a) the velocity of the particle when
x 20 m, (b) the time required for the particle to come to rest.
SOLUTION
PROBLEM 11.25
The acceleration of a particle is defined by the relation 2.5,akv where k is a constant. The particle starts at
0x with a velocity of 16 mm/s, and when 6x
mm the velocity is observed to be 4 mm/s. Determine
(a) the velocity of the particle when 5x mm, (b) the time at which the velocity of the particle is 9 mm/s.
SOLUTION
PROBLEM 11.25 (Continued)
Integrate using t=0 and v=16 mm/s as the lower limits of the integrals
52
016
-k
tv
dt v dv
Problem 11.26
A human powered vehicle (HPV) team wants to model the
acceleration during the 260 m sprint race (the first 60 m is called a
flying start) using a = A – Cv
2
, where a is acceleration in m/s
2
and v
is the velocity in m/s. From wind tunnel testing, they found that
C = 0.0012 m
-1
. Knowing that the cyclist is going 100 km/h at the
260 meter mark, what is the value of A?
SOLUTION
PROBLEM 11.27
Experimental data indicate that in a region downstream of a given
louvered supply vent the velocity of the emitted air is defined by
0
0.18 / ,vvx where v and x are expressed in m/s and meters,
respectively, and 0
v is the initial discharge velocity of the air. For
03.6 m/s,v determine (a) the acceleration of the air at 2 m,x
(b) the time required for the air to flow from 1
x
to x3 m.
SOLUTION
dv
PROBLEM 11.28
Based on observations, the speed of a jogger can be approximated by the relation
0.3
7.5(1 0.04 ) ,vx where v and x are expressed in mi/h and miles, respectively.
Knowing that 0x at 0,t determine (a) the distance the jogger has run when 1 h,t
(b) the jogger’s acceleration in ft/s2 at t
0, (c) the time required for the jogger to run 6 mi.
SOLUTION
dx vx
PROBLEM 11.29
The acceleration due to gravity at an altitude y above the surface of the earth can be
expressed as
62
32.2
[1 ( / 20.9 10 )]
ay
where a and y are expressed in ft/s2 and feet, respectively. Using this expression, compute
the height reached by a projectile fired vertically upward from the surface of the earth if
its initial velocity is (a) 1800 ft/s, (b) 3000 ft/s, (c) 36,700 ft/s.
SOLUTION
We have
2
32.2
dv
va
Then
max
0
0
2
0
32.2
y
vy
vdv dy
PROBLEM 11.29 (Continued)
2
10
(36,700) 3.03 10 ft
PROBLEM 11.30
The acceleration due to gravity of a particle falling toward the earth is 22
/,agRr where r
is the distance from the center of the earth to the particle, R is the radius of the earth, and g
is the acceleration due to gravity at the surface of the earth. If 3960 mi,Rcalculate the
escape velocity, that is, the minimum velocity with which a particle must be projected
vertically upward from the surface of the earth if it is not to return to the earth. (Hint:
0v for .)r
e
PROBLEM 11.31
The velocity of a particle is 0[1 sin ( / )].vv tT
Knowing that the particle starts from the origin with an
initial velocity 0,v determine (a) its position and its acceleration at 3,tT
(b) its average velocity during the
interval 0t to .tT
ave
0
Problem 11.32
An eccentric circular cam, which serves a similar function as the
Scotch yoke mechanism in Problem 11.13, is used in
conjunction with a flat face follower to control motion in pumps
and in steam engine valves. Knowing that the eccentricity is
denoted by e, the maximum range of the displacement of the
follower is dmax, and the maximum velocity of the follower is
vmax, determine the displacement, velocity, and acceleration of
the follower.
max
2
Problem 11.33
An airplane begins its take-off run at A with zero
velocity and a constant acceleration a. Knowing that
it becomes airborne 30 s later at B and that the
distance AB is 900 m, determine (a) the acceleration
a, (b) the take-off velocity B
v.
B
PROBLEM 11.34
A motorist is traveling at 54 km/h when she
observes that a traffic light 240 m ahead of her
turns red. The traffic light is timed to stay red for
24 s. If the motorist wishes to pass the light
without stopping just as it turns green again,
determine (a) the required uniform deceleration of
the car, (b) the speed of the car as it passes the
light.
SOLUTION
Problem 11.35
Steep safety ramps are built beside mountain highways to
enable vehicles with defective brakes to stop safely. A truck
enters a 750-ft ramp at a high speed 0
v and travels 540 ft in 6 s
at constant deceleration before its speed is reduced to 0/2.v
Assuming the same constant deceleration, determine (a) the
additional time required for the truck to stop, (b) the additional
distance traveled by the truck.
BA
