DYNAMICS  THEORY



If an object moves in a curved path, it is said to be in curvilinear motion. It is useful to use vectors to describe the motion of a curved path. 





Position

Position Vector 

To define the position of a point P, select a fixed coordinate system with an origin O. The position of the point P is defined by a position vector, r, connecting O to P.
If P is in motion relative to O, then the position vector will change with time,




Position and Velocity


Average Velocity


Here, r(t) denotes the position of P at time t. If r(t
+ Δt) denotes the position at time t + Δt,
then Δr represents the change in
position,
Δr =
r(t + Δt)  r(t)
The average velocity is the overall change of position over the overall
change of time,
Average velocity = Δr /
Δt 



Instantaneous Velocity


Instantaneous Velocity


Usually, instantaneous velocity is needed, not average velocity. To determine the instantaneous velocity, take the average velocity over
smaller and smaller increments of time,




Acceleration


Acceleration


If v(t) denotes the velocity at time t and v(t + Δt)
denotes the velocity at time t + Δt, then
Δv =
v(t + Δt)  v(t)
Average acceleration = Δv/Δt




Instantaneous Acceleration 

Instantaneous Acceleration


To determine the instantaneous acceleration, take the average acceleration over smaller and smaller increments of time, giving
a = dv/dt






Rectangular Coordinates

Rectangular Coordinate System 

Let the origin O reside at the origin of the Cartesian axis. Resolve
the position vector r(t) into rectangular x, y, and z components,
r = xi + yj + zk 



Velocity


Differentiate to find velocity, giving






Acceleration



To find the acceleration, take a derivative of the velocity, which gives,
Rectangular motion is useful if a_{x} depends only on t, x, and/or v_{x}
(likewise for a_{y} and a_{z}).
Note that the equations describing the motion in each coordinate direction are identical to those of linear motion, as denoted in the Position, Velocity, Acceleration section. 





2D Projectile Problems



Most problems involve projectiles with constant gravity acceleration in only the y direction. In this special case, the rectilinear motion equations become (assumes up is positive y), 




y direction (gravity) 

x direction (no gravity) 

v_{y}(t) = v_{yo}  g (t  t_{o}) 

v_{x}(t) = v_{xo} 
y(t) = y_{o} + v_{yo} (t  t_{o})  g (t  t_{o})^{2} / 2 

x(t) = x_{o} + v_{xo} (t  t_{o}) 
v_{y}^{2} = v_{o}^{2} 
2 g_{} (x  x_{o}) 

v_{x}^{2} = v_{xo}^{2} 



