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CHAPTER 11
PROBLEM 11.1
A snowboarder starts from rest at the top of a double black diamond hill. As he rides down the slope, GPS
coordinates are used to determine his displacement as a function of time: x= 0.5t3 + t2 + 2t where x and t are
expressed in ft and seconds, respectively. Determine the position, velocity, and acceleration of the boarder
when t = 5 seconds.
SOLUTION
PROBLEM 11.2
The motion of a particle is defined by the relation 32
2 9 12 10,xt t t where x and t are expressed in feet
and seconds, respectively. Determine the time, the position, and the acceleration of the particle when v 0.
SOLUTION
PROBLEM 11.3
The vertical motion of mass A is defined by the relation 10 sin 2 15cos 2 100,xtt
where x and t are expressed in mm and seconds, respectively. Determine (a) the position,
velocity and acceleration of A when t 1 s, (b) the maximum velocity and acceleration of A.
SOLUTION
10sin 2 15cos 2 100
xtt
20cos 2 30sin 2
dx
vtt
dt
40sin 2 60cos 2
dv
att
dt
PROBLEM 11.4
A loaded railroad car is rolling at a constant velocity
when it couples with a spring and dashpot bumper
system. After the coupling, the motion of the car is
defined by the relation
4.8
60 sin16
t
x
et
where x and t
are expressed in mm and seconds, respectively.
Determine the position, the velocity and the
acceleration of the railroad car when (a) t 0,
(b) t 0.3 s.
SOLUTION
4.8
t
PROBLEM 11.5
The motion of a particle is defined by the relation 43 2
621233,xt t t t
where x and t are expressed in
meters and seconds, respectively. Determine the time, the position, and the velocity when 0.a
2/3
33 3
2/3
PROBLEM 11.6
The motion of a particle is defined by the relation
32
9248,xt t t
where x and t are expressed in inches
and seconds, respectively. Determine (a) when the velocity is zero, (b) the position and the total distance
traveled when the acceleration is zero.
SOLUTION
32
PROBLEM 11.7
A girl operates a radio-controlled model car in a vacant
parking lot. The girl’s position is at the origin of the xy
coordinate axes, and the surface of the parking lot lies in the
x-y plane. She drives the car in a straight line so that the x
coordinate is defined by the relation x(t) = 0.5t3 - 3t2 + 3t +
2, where x and t are expressed in meters and seconds,
respectively. Determine (a) when the velocity is zero, (b)
the position and total distance travelled when the
acceleration is zero.
SOLUTION
32
PROBLEM 11.8
The motion of a particle is defined by the relation
3
22,xt t where x and t are expressed in feet and
seconds, respectively. Determine (a) the two positions at which the velocity is zero, (b) the total distance
traveled by the particle from t 0 to t 4 s..
SOLUTION
3
22xt t t
PROBLEM 11.9
The brakes of a car are applied, causing it to slow down at a rate
of 10 ft/s2. Knowing that the car stops in 100 ft, determine
(a) how fast the car was traveling immediately before the brakes
were applied, (b) the time required for the car to stop.
SOLUTION
0
(b) Time to stop.
0
0
0
0
0
10
10
010
77.5
10 10
f
t
v
f
f
dv a
dx
dv dt
vt
v
f
PROBLEM 11.10
The acceleration of a particle is defined by the relation 0.2
3,
t
ae
where a and t are expressed in 2
ft/s and
seconds, respectively. Knowing that x 0 and v 0 at t 0, determine the velocity and position of the
particle when t 0.5 s.
SOLUTION
t
PROBLEM 11.11
The acceleration of a particle is directly proportional to the square of the time t. When 0,t the particle is
at 24 m.x Knowing that at 6 s, 96 mtx
and 18 m/s,v
express x and v in terms of t.
27
PROBLEM 11.12
The acceleration of a particle is defined by the relation 2.akt (a) Knowing that 8 m/s
v when t 0
and that 8 m/sv when 2 s,tdetermine the constant k. (b) Write the equations of motion, knowing also
that 0x when 2 s.t
2
2
PROBLEM 11.13
A Scotch yoke is a mechanism that transforms the circular motion of a
crank into the reciprocating motion of a shaft (or vice versa). It has been
used in a number of different internal combustion engines and in control
valves. In the Scotch yoke shown, the acceleration of Point A is defined by
the relation a 1.8sin kt, where a and t are expressed in m/s2 and seconds,
respectively, and k 3 rad/s. Knowing that x 0 and v 0.6 m/s when t 0,
determine the velocity and position of Point A when t 0.5 s.
SOLUTION
PROBLEM 11.14
For the scotch yoke mechanism shown, the acceleration of Point A is defined by the
relation 1.08sin 1.44cos ,aktkt where a and t are expressed in m/s2 and
seconds, respectively, and k = 3 rad/s. Knowing that x = 0.16 m and v = 0.36 m/s
when t = 0, determine the velocity and position of Point A when t = 0.5 s.
SOLUTION
PROBLEM 11.15
A piece of electronic equipment that is surrounded by
packing material is dropped so that it hits the ground with a
speed of 4 m/s. After contact the equipment experiences an
acceleration of
,akx
where k is a constant and x is the
compression of the packing material. If the packing material
experiences a maximum compression of 20 mm, determine
the maximum acceleration of the equipment.
PROBLEM 11.16
A projectile enters a resisting medium at 0x
with an initial velocity
v0900 ft/s and travels 4 in. before coming to rest. Assuming that the
velocity of the projectile is defined by the relation 0,vv kx where v is
expressed in ft/s and x is in feet, determine (a) the initial acceleration of the
projectile, (b) the time required for the projectile to penetrate 3.9 in. into the
resisting medium.
SOLUTION
First note
PROBLEM 11.17
The acceleration of a particle is defined by the relation /.akx
It has been experimentally determined that
15 ft/sv when 0.6 ftx and that 9 ft/sv
when 1.2 ft.x
Determine (a) the velocity of the particle
when 1.5 ft,x (b) the position of the particle at which its velocity is zero.
SOLUTION
PROBLEM 11.18
A brass (nonmagnetic) block A and a steel magnet B are in equilibrium in a brass
tube under the magnetic repelling force of another steel magnet C located at a
distance x 0.004 m from B. The force is inversely proportional to the square of
the distance between B and C. If block A is suddenly removed, the acceleration
of block B is 2
9.81 / ,akx where a and x are expressed in m/s2 and m,
respectively, and 432
410 m/s.k
Determine the maximum velocity and
acceleration of B.
SOLUTION
k
2
m
m
