ME 4590 Dynamics of Machinery
Curvilinear Motion Review
General Concepts
Position, Velocity, and Acceleration
If a particle does not move in a straight line,
then its motion is said to be curvilinear. Given
r (t ) the position vector of a particle P, the
velocity and acceleration of P are defined to be
v
dr
dt
and
a
dv
.
dt
The velocity v is tangent to the path of P at all
times. The acceleration a is not usually tangent
to the path.
Cartesian Coordinates
If we use Cartesian coordinates, then the
position, velocity, and acceleration vectors may
be written as
r  t   x t  i  y t  j  z t  k
v t   x t  i  y t  j  z t  k
a t   x t  i  y t  j  z t  k
Note that the methods for straight-line motion can be applied in each direction.
Kamman – ME 4590: page 1/2
Example: The Projectile Problem
If we neglect air resistance, the motion of a projectile can be analyzed using the
equations for constant acceleration. The horizontal motion (X-direction) occurs at a
constant velocity, and the vertical motion (Y-direction) occurs at a constant acceleration.
The equations that apply in the X and Y directions are
X-direction: (constant velocity, v x0 )
xt   x0  v x0 t
Y-direction: (constant acceleration,  g )
v y t   v y0  gt
v 2y  v 2y  2 g  y  y0 
0
y  t   y0  v y0 t  gt 2
1
2
Derivative of a Rotating Unit Vector
Given the unit vector
e  cos   i  sin   j ,
we can differentiate with respect to time to get


e   sin   i   cos   j    sin   i  cos   j   e
But, we also note that e  k  e , so that we can write
 
e    k  e   k  e    e
Here,  is the angular velocity of the unit vector set  e, e  .
Kamman – ME 4590: page 2/2