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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 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)

, , z

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Plane Rectilinear Motion

◆ What is plane motion? 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.

' r r r

's s s

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

interval .t

avg t

rv

0limt

dt dt

r 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( ) TS

sp tv

avg

stv

Average velocity

t

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

the time interval .t

Instantaneous acceleration is defined as time derivatives of velocity

avg t

va

2

20limt

d dt dt dt

v 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

2 20 02 ( )cv v a s s

0 0 0

0

c

c

v t t

c cv

c

dva constdt

dv a dt

dv a dt a dt

v v a t

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

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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)