12 2. KINEMATICS OF PARTICLES
2.3 UNIFORM AND UNIFORMLY ACCELERATED
RECTILINEAR MOTION
Two types of rectilinear motion are often met in the physical world. ey are the uniform recti-
linear motion and the uniformly accelerated rectilinear motion.
In the uniform rectilinear motion the velocity v of the particle is constant
dx
dt
D v D constant
such that
dx D v dt
and upon integrating w.r.t. t, one has
x x
o
D vt or x D x
o
C vt (2.7)
in which the subscript denotes the initial condition.
In the uniformly accelerated rectilinear motion the acceleration a of the particle is constant
such that upon integrating w.r.t. t once as in the following:
dv
dt
D a D constant or dv D a dt
v v
o
D at or v D v
o
C at (2.8)
and upon integrating w.r.t. t again (starting from
dx
dt
D v such that dx D v dt)
x
Z
x
o
dx D
t
Z
0
v dt D
t
Z
0
.v
o
C at / dt D v
o
t C
1
2
at
2
to give
x D x
o
C v
o
t C
1
2
at
2
: (2.9)
If Equation (2.3) is applied,
dv
dx
v D a D constant
such that
v dv D a dx
and upon integrating as well as rearranging, one has
v
2
D v
2
o
C 2a
.
x x
o
/
: (2.10)
2.3. UNIFORM AND UNIFORMLY ACCELERATED RECTILINEAR MOTION 13
Equations (2.8), (2.9), and (2.10) are the basic equations for uniformly accelerated rectilinear
motion.
e concept of relative position coordinate of two or more particles may be applied to
cases in which particles or blocks (conceptually viewed as particles) are connected by inextensible
cords or cables. For example, the blocks shown in Figure 2.4 has the linear relation between their
position coordinates as
x
A
C 2x
B
D constant: (2.11)
is is known as the geometric or holonomic constraint in more advanced textbook. First and
second derivatives w.r.t. t can be performed to obtain the relative velocity and acceleration.
x
A
x
B
A
B
Figure 2.4: Blocks connected by inextensible cables.
Remarks:
In choosing the distances between points or blocks (conceptually regarded as points here) in this
type of problems a question is often asked: Why the radius of the wheel or pulley or length of
the arm that is fixed to the ground or reference positions is disregarded? e answer is that since
these radius and fixtures do not change with time and therefore they can be considered as part
of the constant being used in the expression or equation of position.
Example 2.4
Slider block A travels to the left, in Figure 2.5, with a constant velocity of 8 m/s and is connected
by an inextensible cable to block B. It is assumed that no friction is present between the block
and the supporting horizontal surface, and between the cable and pulley. Determine
(a) the velocity of block B,
(b) the velocity of portion D of the cable, and
14 2. KINEMATICS OF PARTICLES
A
B
C
D
x
A
y
B
y
C
y
D
+
+
Figure 2.5: Blocks travelling horizontally and vertically.
(c) the relative velocity of portion C of the cable with respect to portion D.
Solution:
Let x and y be the horizontal and vertical distances, respectively. Assuming positive distances
as shown in Figure 2.5, one has
x
A
C 3y
B
D constant:
en, the velocity equation, by taking the time derivative of this position equation, becomes
v
A
C 3v
B
D 0; (2.12a)
taking the time derivative again, one obtains
a
A
C 3a
B
D 0: (2.12b)
(a) Substituting v
A
D 8 m/s into Equation (2.12a), one has
v
B
D
8
3
m/s or
*
v
B
D
8
3
m/s " :
(b) With reference to Figure 2.5,
y
B
C y
D
D constant
2.3. UNIFORM AND UNIFORMLY ACCELERATED RECTILINEAR MOTION 15
such that
v
B
C v
D
D 0:
is gives
v
D
D
8
3
m/s or
*
v
D
D
8
3
m/s # :
(c) With reference to Figure 2.5,
x
A
C y
C
D constant
such that
v
A
C v
C
D 0 ) v
C
D v
A
D 8 m/s:
is gives the relative velocity
v
C=D
D v
C
v
D
D
8
8
3
m/s or
*
v
C=D
D 10
2
3
m/s " :
Example 2.5
Collar A travels upward from rest with a constant acceleration, as shown in Figure 2.6. Note
that no friction is present between the collars and the vertical columns. e cable is inextensible.
ere is no friction between the cable and pulleys. Knowing that after 6 s the relative velocity
of collar B respect to collar A is 20 m/s, determine
(a) the accelerations of blocks A and B, and
(b) the velocity of block B after 6 s.
Solution:
Let y be positive downward. With reference to Figure 2.6,
2y
A
C y
B
C
.
y
B
y
A
/
D constant:
en, the velocity equation, by taking the time derivative of this position equation, becomes
v
A
C 2v
B
D 0; (2.13a)
taking the time derivative again, one obtains
a
A
C 2a
B
D 0: (2.13b)
16 2. KINEMATICS OF PARTICLES
y
A
y
C
y
B
+
A
B
C
Figure 2.6: Collars traveling vertically.
(a) Applying Equation (2.13a) for the initial condition when collar A starts from rest such
that
.
v
A
/
o
C 2
.
v
B
/
o
D 0 or
.
v
B
/
o
D 0:
Since collar A moves upward therefore it is constant and negative. is leads, by applying
Equation (2.13b), to a
B
being positive or moving downward. us, upon integrating w.r.t.
t of the acceleration term individually,
v
A
D
.
v
A
/
o
C a
A
t ) v
A
D a
A
t
and
v
B
D
.
v
B
/
o
C a
B
t ) v
B
D a
B
t
because .v
A
/
o
D 0; .v
B
/
o
D 0 as the system starts from rest. But, the relative velocity,
v
B=A
D v
B
v
A
D
.
a
B
a
A
/
t:
From Equation (2.13b), a
B
D
a
A
2
so that
v
B=A
D
3
2
a
A
t ) 20 D
3
2
a
A
.6/ ) a
A
D
20
9
m/s
2
and
a
B
D
10
9
m/s
2
:
at is,
*
a
A
D
20
9
m/s
2
" and
*
a
B
D
10
9
m/s
2
# :
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