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 freefall acceleration directed downward
Kinematic equations for x and y components of position vector:
The x and y components of velocity:
