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.