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# 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: