Curvilinear Kinematics
SI units & Physics constants
Curvilinear Kinematics investigates lows of motion of objects in space in two and three directions without any reference to forces that cause the motion to change

Here (all units see here):
is original time on time interval 
is final time
is position vector at time 
is position vector at time 
is displacement vector during time 
is length of path during time 
is mean velocity vector during time 
is instantaneous velocity vector at time , tangent to trajectory
is total acceleration vector at time 
are unit vectors of axes x, y, z respectively
General formulas
Displacement vector during time is defined by vector difference
Average velocity vector during time 

Instantaneous velocity vector

Average speed during time 
Instantaneous speed

Total instantaneous acceleration vector

Relation between acceleration and velocity vectors

Kinematics equation for motion

Relation between curviliner and rectilinear kinematics is defined by vector sum

where:
x(t), y(t), z(t) are coordinates of position vector as functions of time t, defined by rectilinear kinematics
are unit vectors of axes x, y, z shown in the above diagram
Components of total acceleration vector

Where:
R is radius of curvature of trajectory at time 
is centripetal acceleration, normal to trajectory and directed to its center
is tangential acceleration, tangent to trajectory and parallel to velocity
is total acceleration vector
Magnitude of total acceleration

Angle between vectors and 

Projectile motion

Where:
is original velocity
is original angle
is free-fall acceleration directed downward
Kinematic equations for x and y components of position vector:


The x and y components of velocity:


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