# Kinematics of a Particle

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Kinematics of a ParticleKaiser Rex N. Pama

Engineering Mechanics- Dynamics

Rectilinear Kinematics: Continuous Motion

The Kinematics of a particle is characterised by specifying, at any given instant, the particles

position, velocity, and acceleration.

Position and Displacement

O ss

Position

O s

Displacement

s

ss

s=s'-s

Velocity

vav =st

Os

sv

v = dsdt+

(vsp )avg =sT t

Accelerationaav =

v t a =

dvdt

Os

v v'

a

Os

v v'

aP P

Deceleration

v = dsdt a =dvdt

dt = dsvdt = dvadsv =

dva

ads = vdv

Constant accelerationVelocity as a function of time

Position as a function of time

Velocity as a function of position

dvv0

v

= ac dt0

t

v = v0 + act

dss0

s

= (v0 + act)dt0

t

s = s0 + v0t +12 act

2

dvv0

v

= ac dss0

s

v2 = v02 + 2ac(s s0 )

dvv0

v

= ac dt0

t

v = v0 + act

dss0

s

= (v0 + act)dt0

t

s = s0 + v0t +12 act

2

dvv0

v

= ac dss0

s

v2 = v02 + 2ac(s s0 )

dvv0

v

= ac dt0

t

v = v0 + act

dss0

s

= (v0 + act)dt0

t

s = s0 + v0t +12 act

2

dvv0

v

= ac dss0

s

v2 = v02 + 2ac(s s0 )

a = ac

Rectilinear Kinematicsv = dxdt a =

dvdt

Curvilinear MotionPosition

The position is designated by r = r(t).

Displacement

the displacement is r = r - r

Velocity

Average velocity is: vavg = r/ t . Instantaneous velocity is: v = dr/dt . v is always tangent to the path

What is speed v? s r as t0, then v = ds/dt.

Acceleration

Average acceleration is: aavg = v/t = (v - v)/t

Instantaneous acceleration is: a = dv/dt = d2r/dt2

Position

The position can be defined as r = x i + y j + z k where x = x(t), y = y(t), and z = z(t) .

Magnitude is: r = (x2 + y2 + z2)0.5 Direction is defined by the unit vector: ur = (1/r)r

Curvilinear Motion: Rectangular Components

Velocity

Acceleration

Motion of a Projectile

Horizontal Motion

Vertical Motion

Curvilinear Motion: Normal and Tangential Components

Normal (n) and tangential (t) coordinates are used when a particle moves along a curved path and the path of motion is known

n and t directions are defined by the unit vectors un and ut, respectively.

Position

Radius of curvature, , is the perpendicular distance from the curve to the center of curvature at that point.

The position is the distance, s, along the curve from a fixed reference point.

Radius of

Curvature

The velocity vector is always tangent to the path of motion (t-direction).

The magnitude is determined by taking the time derivative of the path function, s(t).

where

Velocity

Acceleration is a = dv/dt = d(vut)/dt = vut + vut

. .

Here v represents the change in the magnitude of velocity and ut change in the direction of ut.

..

.

The acceleration vector can be expressed as:

where

or

Acceleration

SPECIAL CASES OF MOTION

There are some special cases of motion to consider.

2) The particle moves along a curve at constant speed. at = v = 0 => a = an = v2/

.

The normal component represents the time rate of change in the direction of the velocity.

1) The particle moves along a straight line. => an = v2/ = 0 => a = at = v

.

The tangential component represents the time rate of change in the magnitude of the velocity.

MOTION RELATIVE TO A FRAME IN TRANSLATION

relative position of B with respect to A

relative velocity of B with respect to A

relative acceleration of B with respect to A

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