
Curvilinear Motion
◦ It occurs when a
particle moves along a
curved path

Position
◦ The position of a
particle on a space
curve will be designated
by the position vector, r
= r(t)
◦ It’s a function of time
and change magnitude
and direction as it
moves along the curve
General Curvilinear Motion

Displacement
◦ Represent the change
in the particle’s
position and is
determine by vector
subtraction
r  r - r
General Curvilinear Motion

Velocity
◦ During the time interval
Δt, the average velocity
r
v avg 
t
◦ With smaller value of t ,
the instantaneous velocity
dr
v
dt
The speed,
ds
v
dt
General Curvilinear Motion

Acceleration
◦ Provided the velocity of
the particle is known at
the two points P and P’
during the time interval
Δt,
a avg
v

t
◦ With smaller value of t ,
the instantaneous
acceleration
dv d 2r
a
 2
dt dt
General Curvilinear Motion


The motion can be
describe along a path that
can be represented using
a fixed x, y, z frame of
reference
Position
◦ The position of a
particle P at a point
(x, y, z) on the curve
s, is defined by the
position vector
r  xi  yj  zk
r  x2  y 2  z 2
u r  r/r
Curvilinear Motion: Rectangular
•Position
dr d
d
d
v
 ( xi)   yj  zk 
dt dt
dt
dt
d
xi   dx i  x di
dt
dt
dt
dr
v
 vx i  v y j  vz k
dt
vx  x v y  y vz  z
v  vx2  y y2  z z2
u v  v/v
Curvilinear Motion: Rectangular
•Acceleration
dv
a
 ax i  a y j  az k
dt
ax  vx  x
a y  v y  y
az  vz  z
a  a a a
2
x
2
y
2
z
u a  a/a
Curvilinear Motion: Rectangular

The free-flight motion of a projectile is often
studied in terms of its rectangular components,
since the projectile’s acceleration always acts
in the vertical direction
Motion of a Projectile

Horizontal Motion

Vertical Motion
ay  g
ax  0
vx   v0 x
x  x0  v0 x t
v y  v0 y  act
y  y0  v0 y t  12 act 2
v  v
2
y

2
0 y
 2ac  y  y0 


When the path along
which a particle is moving
is known, then it is often
convenient to describe the
motion using n and t
coordinate which acts
normal and tangential to
the path
Planar Motion
◦ Tangential component
 Unit vector, ut
◦ Normal component
 Unit vector, un
CM: Normal and Tangential

Velocity
◦ The particle’s velocity
has a direction that is
always tangent to the
path and a magnitude
that is determine by
taking the time
derivative of the path
function s=s(t)
v  vu t
where,
v  s
CM: Normal and Tangential

Acceleration
◦ The particle’s
acceleration is the
time rate of change
of the velocity
a  v  v ut  v u t
where,
s
v

u t   un  un  u n


CM: Normal and Tangential

Acceleration
a  at ut  an u n
where,
at  v at ds  vdv
an 
a
v2

a
2
t
a
2
n

CM: Normal and Tangential


Some engineering
problems involve angular
position and a radial
distance
Polar coordinates
◦ Radial, r (ur)
◦ Transverse, θ (uθ)

Position
r  vu r
CM: Cylindrical

Velocity
v  r  rur  r u r
where,
u r 
 
u r  lim
  lim
u
t 0 t
 t 0 t 
u r  u
v  vr u r  v u
vr  r v  r
CM: Cylindrical

Acceleration
a  v  rur  r u r  r u  ru  r u 
u
 

  lim
u r
t 0 t
 t 0 t 
u   u r
where, u   lim
a  ar u r  a u
ar  r  r 2
a
a
a  r  2r
2
r
 a2

CM: Normal and Tangential

Cylindrical coordinates
◦ Radial, r (ur)
◦ Transverse, θ (uθ)
◦ Altitude, z (uz)

Position, velocity and
acceleration
rp  r u r  z u z
v  r u r  r u  zu z
2



a  r  r u r  r  2r u  zu z

 
CM: Cylindrical
