Chapter I:
KINEMATICS OF PARTICLES
A particle is a body whose physical dimensions are very
small compared with the radius of curvature of its path.
The figure shows a particle moving along some general
POSITION, VELOCITY, ACCELERATION
curvilinear path in
space and is located at
point P at a certain
instant.
path k
i
OPr /
j
P
O
z
z
y y x x
kzjyixr OP
/
The position vector of the particle at any time t
can be described by fixed rectangular coordinates
measured from the
origin O to point P
as path
k
i
OPr /
j
P
O
z
z
y y x x
Let us construct a coordinate system x'y'z', parallel to
the xyz coordinate system and let the origin of this
system be point O'.
The position vector of
point P with respect
to O' is
kzjyixr OP
/
path
OPr /
P
O
z
y x
z'
x' y'
OOr /
OPr /
O'
According to the parallelogram law of
addition, the relation between the position
vectors is:
OPOOOP rrr ///
path
OPr /
P
O
z
y x
z'
x' y'
OOr /
OPr /
O'
In both equations is a fixed vector. The difference between the position vectors of a body at two different instants is named “displacement” (yer değiştirme). At time t if the body is at point P and at time t+dt it is at point Q, then the displacement is,
For point Q OQOOOQ rrr ///
// OPOQ rrr
d
OOr /
rd
OP/OQ/ rrrδ
or rd
path
OPr /
P
O
z
y x
z'
x' y'
Q O' OPr /
OQr /
OOr /
t
tt d
OQr /
This displacement is independent of the origin of the
coordinate system since
For an instantaneous
time difference the
direction of will
be tangent to the
path.
rd
rd
path
OPr /
P
O
z
y x
z'
x' y'
Q O' OPr /
OQr /
OOr /
t
tt d
OQr /
- - //////
//
OPOQ
r
OPOO
r
OQOO rrrrrrr
OPOQ
d
Also the direction of will be the same as the
direction of , that is tangent to the path of the
body.
Since the displacement is independent of the
origin of the coordinate system, the velocity vector
will also be independent of the origin of the
coordinate system.
The velocity of a body is defined as the time rate of
change of its position.
OPOP
tP r
dt
rd
tr
v //
0lim
dd
d
rd
Pv
Pv
rd
The velocity vector in terms of its components,
zyx
zyxP
vdt
dzzv
dt
dyyv
dt
dxx
kzjyixkvjvivv
The acceleration of a body is defined as the time rate
of change of its velocity,
p
pa
dt
vd
The acceleration vector in terms of its
components is given as
2
2
2
2
2
2
dt
dv
dt
zda
dt
dv
dt
yda
dt
dv
dt
xda
kzjyixa
kvjvivkajaiaa
zz
y
y
xx
P
zyxzyxP
If the position, velocity and acceleration of a body can
be described by only their x components, the y and z
components are zero, such a motion is named as
“rectilinear motion” (doğrusal hareket) . In this case,
the x axis may be taken as the axis of the motion and the
body may move along a straight line with varying velocity
and acceleration.
If only the z components of the position, velocity
and acceleration vectors are zero, x and y
components are different from zero, such a motion
is named as “ plane curvilinear motion” (düzlemde
eğrisel hareket) .
If all the components of the position, velocity and
acceleration vectors related to the motion of a body are
different from zero, such a motion is named as “general
curvilinear motion” or “space curvilinear motion”
(genel eğrisel hareket veya uzayda eğrisel hareket).
Consider a particle moving along a straight line which is at
point P at time t. The position of the particle at time t is s
measured from some convenient reference point O fixed
on the line.
svaasvvsr PPOP ,,/
t=0 t t+Dt
v=v0
s=0
At time t+Dt the particle has moved to P’ and its coordinate
becomes s+Ds. The change in the position coordinate during
the interval Dt is called the displacement Ds. The
displacement would be negative if the particle moved in the
negative s-direction.
t=0 t t+Dt
v=v0
s=0
svaasvvsr PPOP ,,/
The average velocity of the particle during the interval Dt
is the displacement divided by the time interval or
t
svav
D
D
As Dt becomes smaller and approaches zero in the limit, the
average velocity approaches the instantaneous velocity of
the particle
st
sv
t
D
D
D dt
ds lim
0
Thus, the velocity is the time rate of change of the
position coordinate s. The velocity is positive or negative
depending on whether the corresponding displacement is
positive or negative. v is (+) if the particle is moving right
and v is (-) if the particle is moving left.
t=0 t t+Dt
v=v0
s=0
The average acceleration of the particle during the
interval Dt is the change in its velocity divided by the
time interval or
t
vaav
D
D
As Dt becomes smaller and approaches zero in the limit,
the average acceleration approaches the instantaneous
acceleration of the particle
svdt
sd
dt
dv
t
va
t
D
D
D 2
2
0 lim
The acceleration is positive or negative depending on
whether the velocity is increasing or decreasing.
If the signs of velocity and acceleration are the same
then velocity is increasing and it is said that the
particle is accelerating (pozitif ivmelenme, hızlanma).
If the signs of the velocity and acceleration are
opposite, then velocity is decreasing and it is said that
the particle is decelerating (negatif ivmelenme,
yavaşlama).
Note that the acceleration would be positive if
the particle had a negative velocity which
becoming less negative.
By eliminating the time dt between velocity and
acceleration equations, we obtain a differential
equation relating, displacement, velocity and
acceleration.
sdsdssorvdvads
vds
dv
dt
dva
v
dsdt
,
The basic relationships between displacement,
velocity, acceleration and time can be summarized as:
sdsdssorvdvads
sv dtds sv
dt
sddtdv
a 2
2
Although position, velocity and acceleration are
vector expressions, for rectilinear motion, where
the direction of the motion is that of the given
straight-line path, they may be expressed by
their magnitudes indicating the sense of the
vectors along the path described by a plus (+) or
minus (-) sign.
The displacement of the particle may not always be
equal to the path taken. Displacement is the vector
difference between the initial and final positions of the
particle along the path. If the particle completes its
motion at exactly the same point it starts its motion
then the displacement is zero. But the path taken will
be the distance it has traveled along the path and will
not be equal to zero.
1 2
3
start
final position
If the position coordinate s is known for all values of
the time t, successive mathematical and graphical
differentiation with respect to t gives the velocity
and acceleration.
In many problems, however, the relationship between
position coordinate and time is unknown, and it must
be determined by successive integration from the
acceleration.
Depending on the nature of the forces, the
acceleration may be a function of time, velocity or
position coordinate, or a combined function of these
quantities.
The procedure for integrating the differential
equation in each case is indicated as follows:
For all cases let the initial conditions
be assumed as, t=0, s=s0 , v=v0.
020
20
20
2
v
v
200
0
00
0
s
s
00
0
v
v
2 2
1
2
1
00
0
0
ssavvssavvdsavdv
adsvdv
attvssdtatvss
vdtdsvdt
ds
atvvatvv
dtadvadt
dv
s
s
t
t
t
)( )(ds
)( )(
)(dv )(
0
0
0
s
s
)(
0
0
0
0
0
v
v
0
0
tt
tgv
tt
t
dttgssdttg vdt
ds
dttfvvdttfvv
dttftfadt
dv
)(
0
v
vf(v)
dvt
dt f(v)
dvvfa
dt
dv
v
v
v
v
f(v)
vdvss
ds f(v)
vdv ds
f(v)
vdvdsvfadsvdv
0
00
)(
0
s
s
This result gives t as a function of v.
Another approach is
)(
)(
)(2
)(2
)( )(
0
0
0
0
00
0
)(
s
s
20
2
s
s
20
2
s
s
s
s
ts
s
sgv
v
v
g(s)
dst
dtg(s)
dsdt
sg
dssg
dt
dsv
dssfvv
dssfvv
dssf vdv dssfadsvdv
Figure a is a schematic plot of the variation of s with t from time t1 to time t2 for some given rectilinear motion. By constructing the tangent to the curve at any time t, we obtain the slope, which is the velocity v=ds/dt (Figure b). Similarly, the slope dv/dt of the v-t curve at any instant gives the acceleration at that instant (Figure c). a
dt
dv
vdt
dx
vdt
ds
vdt
dx
12
2
1
2
1
ssvdtds
t
t
s
s
(area under v-t curve)
We see from Figure b that the area under the v-t curve during time dt is vdt, which is the displacement ds. The net displacement of the particle during the interval from t1 to t2 is the corresponding area under the curve, which is
Similarly, from Figure c we see that the area under the a-t curve during time dt is adt, which is the velocity dv. The net change in velocity between t1 and t2 is the corresponding area under the curve, which is
12
2
1
2
1
vvadtdv
t
t
v
v
(area under a-t curve)
21
22
2
12
1
2
1
vvadsvdv
s
s
v
v
(area under a-s curve)
When the acceleration a is plotted as a function of the position coordinate s, the area under the curve during a displacement ds is ads is
Example: The slope of the v-t graphic gives the magnitude of the acceleration for that instant.
adt
dv
If the magnitude of the acceleration is (+) then either the velocity is increasing along the +s direction or the velocity is decreasing along –s direction. If acceleration is (–) either the velocity is decreasing along the +s direction or it is increasing along the –s direction.
The slope of the x-t graphic gives the magnitude of the velocity for that instant.
vdt
dx
If v is + the particle moves in +s direction, it v is – the particle moves in –s direction.