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Kinematics of a Particle

◆ What is kinematics?Kinematics: study of motion without reference to the forces which cause motion.

◆ What is motion of a particle?Movement of a particle à need position changes

Thus, when we call motion, it is related to position change, velocity, acceleration.

◆ How to describe the motion of a particle?- Need reference frame and coordinate systems- Coordinate systems

Cartesian coordinates (x, y, z)Spherical coordinates (R, )Cylindrical coordinates (r, )Normal-Tangential coordinates (n-t)

,q f, zq

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Plane Rectilinear Motion

◆ What is plane motion? à a particle moves in a single plane.

◆ Type of motionLectilinear motion Curvilinear motion

Position, velocity, accelerations are vector quantities. However, sense is important for rectilinear motion, because direction is fixed.

Position vector r is used to specify the location of the particle P at any given instant.

Note: r is always along the s axis,direction is never changed

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Rectilinear Motion (displacement)

Displacement of the particle is defined as the change in its position.

'D = -r r r

's s sD = -

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Rectilinear Motion(Velocity)Average velocity is defined as a displacement from P to P’ during the time

interval .tD

avg tD

ºDrv

0limt

dt dtD ®

Dº =

Dr rv

Instantaneous velocity is defined as time derivatives of position

Average speed is defined as the total distance traveled by a particle during the time interval .

TS

avg( ) TSsp tv =

D avg

stv -D

=D

Average velocity

tD

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Rectilinear Motion (Acceleration)Average acceleration is defined as velocity changes from P to P’ during

the time interval .tD

Instantaneous acceleration is defined as time derivatives of velocity

avg tD

=Dva

2

20limt

d dt dt dtD ®

D= = =

Dv v ra

2

2

dv d sadt dt

= =

For rectilinear motiondsv dsdt

dva dvdt

= = vdv ads=

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Constant acceleration

0 0 0

2 20 0

1 ( ) ( )2

cv s s

c cv s s

c

vdv a ds

vdv a ds a ds

v v a s s

=

= =

- = -

ò ò ò

2 20 02 ( )cv v a s s= + -

0 0 0

0

c

cv t t

c cv

c

dva constdt

dv a dt

dv a dt a dt

v v a t

= =

=

= =

- =

ò ò ò

0 cv v a t= +2

0 012 cs s v t a t= + +

0

0

0

00

20 0

( )

( )

12

c

cs t

cs

c

ds v v a tdtds v a t dt

ds v a t dt

s s v t a t

= = +

= +

= +

- = +

ò ò

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Erratic Motion(1)

Slope of s-t graph

= velocity

Given the s-t graph, construct the v-t graph

dsvdt

=

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Erratic Motion(2)Given the v-t graph, construct the a-t graph

Slope of v-t graph

= acceleration

dvadt

=

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Erratic Motion(3)Given the a-t graph,

construct the v-t graphGiven the v-t graph,

construct the s-t graph

displacement= area under v-t graph

Change in velocity

= area under a-t graph

dv adt= ds vdt=

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Erratic Motion(3)Given the a-t graph,

construct the v-t graphGiven the v-t graph,

construct the s-t graph

displacement= area under v-t graph

Change in velocity

= area under a-t graph

dv adt= ds vdt=

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Erratic Motion(4)Given the a-s graph, construct the v-s graph

Given the v-s graph, construct the a-s graph

( )dva vds

=

vdv ads=

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Example (rectilinear motion – constant acceleration)

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Solution (rectilinear motion – constant acceleration)

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Example (rectilinear motion – variable acceleration)

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solution (rectilinear motion – variable acceleration)

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Plane Curvilinear Motion

◆ What is plane motion? à a particle moves in a single plane.

Coordinate system for a plane motion à x-y, (z)Position, velocity, accelerations are vector quantities.We have to consider “direction” and “magnitude” àcurvilinear

motionWe have to consider “sense” and “magnitude” àrectilinear motion

Position vector r measuredfrom some fixed origin

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Plane Curvilinear Motion

◆ Displacement: Vector change of position. Independent of the choice of origin

◆ Distance: Path from one point to the other point.◆ Velocity:

avg tD

=Drv

0limt

dt dtD ®

D= =

Dr rv

0 0

r dslim limdtt t

svt tD ® D ®

D D= = =

D D

Speed=mag. of velocityVelocity is tangent to the path

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Plane Curvilinear Motion

◆ acceleration

avg tD

=Dva

2

20limt

d dt dt dtD ®

D= = =

Dv v ra

Hodograph

Acceleration is tangent to Hodograph

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Curvilinear Motion: Rectangular coordinates

◆ Position= (t) = + +

( ) + ( ) + ( )x y z

x t y t z t=r r i j k

i j k

2 2 2r x y z= + +

Magnitude of position vector

Direction of position vector

r r= ru

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Curvilinear Motion: Rectangular coordinates

◆ Velocity

2 2 2x y zv v v v= + +

Magnitude of velocity vector

Direction of velocity vector

v v= vu

{ } { } { }= (t) = ( ) + ( ) + ( )d d d dx t y t z tdt dt dt dt

dx d dy d dz dx y zdt dt dt dt dt dt

=

= + + + + +

rv v i j k

i j ki j k

x y zdx dy dz v v vdt dt dt

= + + = + +i j k i j k

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Curvilinear Motion: Rectangular coordinates

◆ Acceleration

2 2 2x y za a a a= + +

Magnitude of acceleration vector

Direction of acceleration vector

a a= au

yx zx y z

dvdv dv a a adt dt dtx y z

= + + = + +

= + +

i j k i j k

i j k&& && &&

{ } { } { }= (t) = ( ) + ( ) + ( )x y zd d d dv t v t v tdt dt dt dt

=va a i j k

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Curvilinear Motion: Rectangular coordinates

When is it convenient using Cartesian coordinate systems?X, y components of acceleration are independently generated or determined.

0,x ya a g= = -0x

xdv adt

= = 0( )x xv const v= =

yy

dva g

dt= = -

0( )y yv v gt= -

0( )y ydy v v gtdt

= = - 20 0

1( )2yy y v t gt= + -

0( )xdx vdt

= 0 0( )xx x v t= +

x component: Horizontal motion

y component: Vertical motion

Motion of Projectile

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Example (curvilinear motion – Cartesian coordinates)

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Solution (curvilinear motion – Cartesian coordinates)