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Plane curvilinear motion is the motion of a particle along
a curved path which lies in a single plane.
Before the description of plane curvilinear motion in any
specific set of coordinates, we will use vector analysis to
describe the motion, since the results will be independent
of any particular coordinate system.
At time t the particle is at position A, which is located by the
position vector measured from the fixed origin O. Both the
magnitude and direction of are known at time t. At time
t+Dt, the particle is at A' , located by the position vector .
r
r
rr
D
The displacement of the particle during Dt is the vector which
represents the vector change of position and is independent of the
choice of origin. If another point was selected as the origin the
position vectors would have changed but would remain the
same.
r
D
r
D
The distance actually travelled by the particle as it moves along the
path from A to A' is the scalar length Ds measured along the path. It is
important to distinguish between Ds and .r
D
r
DThe average velocity of the particle between A and A' is defined as
which is a vector whose direction is that of . The magnitude of
is .
The average speed of the particle between A and A' is
Clearly, the magnitude of the average velocity and the speed
approach one another as the interval Dt decreases and A and A'
become closer together.
t
rvav
D
D
r
D
t
s
D
D
t
s
D
D
t
r
D
D
avv
t
r
D
D
The instantaneous velocity of the particle is defined as the limiting value
of the average velocity as the time Dt approaches zero.
We observe that the direction of approaches that of the tangent to the path
as Dt approaches zero and, thus, the velocity is always a vector tangent to
the path.
rdt
rd
t
rlimvt
D
D
D 0
v
r
D
v
sdt
dsvv
The magnitude of is called the speed and is the scalar
The change in velocities, which are tangent to the path and are at A and
at A during time Dt is a vector .
Here indicates both change in magnitude and direction of . Therefore,
when the differential of a vector is to be taken, the changes both in
magnitude and direction must be taken into account.
v
v
v
D
v
D v
which is a vector whose direction is that of . Its magnitude is
The instantaneous acceleration of the particle is defined as the limiting value
of the average acceleration as the time interval approaches zero.
t
vaav
D
D
v
D
rvdt
vd
t
vlimat
D
D
D 0
The average acceleration of the particle
A and A' is defined as
t
v
D
D
The acceleration includes the effects of both the
changes in magnitude and direction of . In
general, the direction of the acceleration of a
particle in curvilinear motion is neither tangent
to the path nor normal to the path.
As Dt becomes smaller and approaches zero, the direction of
approaches .
v
Dvd
v
If the acceleration was divided into two components one tangent and
the other normal to the path, it would be seen that the normal
component would always be directed towards the center of curvature.
If velocity vectors are plotted from some arbitrary point C, a curve,
called the hodograph, is formed. Acceleration vectors are tangent to
the hodograph.
Three different coordinate systems are commonly used in describing
the vector relationships for plane curvilinear motion of a particle.
These are:
• Rectangular (Cartesian) Coordinates
(Kartezyen Koordinatlar)
• Normal and Tangential Coordinates
(Doğal veya Normal-Teğetsel Koordinatlar)
• Polar Coordinates
(Polar Koordinatlar)
The selection of the appropriate reference system is a prerequisite for
the solution of a problem. This selection is carried out by considering
the description of the problem and the manner the data are given.
Cartesian Coordinate system is useful for describing motions where the x-
and y-components of acceleration are independently generated or
determined. Position, velocity and acceleration vectors of the curvilinear
motion is indicated by their x and y components.
As we differentiate with respect to time,
we observe that the time derivatives of
the unit vectors are zero because their
magnitudes and directions remain
constant.
jyixjvivjaiaa
jyixjvivv
jyixr
yxyx
yx
ji
and
Let us assume that at time t the particle is at point A. With the aid of
the unit vectors , we can write the position, velocity and
acceleration vectors in terms of x- and y-components.
The magnitudes of the components of and are:v
a
yvaxva
yvxv
yyxx
yx
In the figure it is seen that the direction of ax is in –x direction.
Therefore when writing in vector form a “-” sign must be added in
front of ax.
x
y
yx
yx
x
y
yx
yx
a
atan
aaa
aaa
v
vtan
vvv
vvv
22
222
22
222
The direction of the velocity is always tangent to the path. No such
thing can be said for acceleration.
If the coordinates x and y are known independently as
functions of time, x=f1(t) and y=f2(t), then for any value of
the time we can obtain .Similarly, we combine their first
derivatives to obtain and their second
derivatives to obtain .
Inversely, if ax and ay are known, then we must take integrals
in order to obtain the components of velocity and position. If
time t is removed between x and y, the equation of the path
can be obtained as y=f(x).
a
v
r
yandx
yandx
Projectile Motion (Eğik Atış Hareketi)
An important application of two-dimensional kinematic theory is the problem
of projectile motion. For a first treatment, we neglect aerodynamic drag and
the curvature and rotation of the earth, and we assume that the altitude change
is small enough so that the acceleration due to the gravity can be considered
constant. With these assumptions, rectangular coordinates are useful to employ
for projectile motion.
Acceleration components;
ax=0
ay= -g
vox= vocos
voy= vosin
vo
g
v
vx
vy
v'
vx
v'y
x
y Apex; vy=0
vx
Horizontal Vertical
)(2
2
1
0
0
2
0
2
2
0000
00
yygvv
gttvyytvxx
gtvvvvconstant,v
gaa
yy
yx
yyxxx
yx
--
-
-
-
We can see that the x- and y-motions are independent of each other.
Elimination of the time t between x- and y-displacement equations
shows the path to be parabolic.
If motion is examined separately in horizontal and vertical directions,
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