Kinematics - Dymanics

Kinematics studies the motion of a particle(object) neglecting the effects of forces.

A particle or object is defined by by its position, velocity, and acceleration.
Velocity is the derivative of the position with respect to time: v = ds/dt
Acceleration is the derivative of the velocity with respect to time: a = dv/dt

Also, relating displacement, velocity and acceleration: a ds = v dv

For constant acceleration:

• position as a function of time: S = S0 + V0t + .5a t^2
• velocity as a function of time: V = V0 + a t
• Velocity as a function of position: V^2 = Vo^2 + 2a(S-S0)

Rectangular Components in Curvilinear Motion

• Position: r = x + y + z
• Velocity: V = Vx + Vy + Vz
• Acceleration: A = ax +ay + az

The magnitude of either the position, velocity, or acceleration is the square root of the sum of the and components squared.
Magnitude = sqrt( ^2 + ^2 + ^2)


Normal & Tangential Components for Curvilinear Motion

Normal (en): unit vector perpendicular to the path and points toward the center of curvature.
Tangential (et): unit vector tangent to the particles path.

Velocity: V = V et
Acceleration: A = At et + An en
where At is the derivative of the velocity, and An is v^2 / ρ
An is also know as centripetal acceleration.
If the radius of curvature(ρ) is not given, using the equation of the particles path...
ρ = [1 + (dy/dx)^2]^(3/2) / (d^2 y / dx^2)


Cylindrical Components for Curvilinear Motion

Also known as polar coordinates. er is a unit vector in the radial direction(points from the origin to a point). eθ is a unit vector that is perpendicular to the radial direction and points in the positive θ direction.

Position: r = r er
Velocity: v = r(dot) er + r θ(dot) eθ
Acceleration: a = r(double dot) - r θ^2(dot) er + r θ (double dot) + 2 r(dot) θ(dot) eθ
The (dot) and (double dot) refers to a dot over the character before it, and represents the derivative or second derivative with respect to time.

• θ(dot) is the angular velocity
• θ(double dot) is the angular acceleration

Projectile Motion

After a particle has been launched, its weight is the only force acting on it, neglecting air resistance. This weight(due to gravity) will cause a constant acceleration downward.

Horizontal motion:

• Position: X = X0 + V0 X t
• Velocity: V = V0 cos(θ) this will be constant when drag is not considered
  

Vertical motion:

• Position: Y = Y0 + V0 t - .5gt^2
• Velocity: V = V0 sin(θ) - gt

Also, relating position, velocity and angle of launch:
Y = (Tan(θ))X [g/2V0^2 (cos(θ)^2)] X^2


For more topics on Dynamics click here