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,