Kinematics

Kinematics is the branch of mechanics concerned with the description of motion without regard to the forces that cause said motion. For a gentle introduction see an intro to kinematics.

Position Vectors

The position of a point in three-dimensional space is described by a position vector measured from a chosen origin . In a fixed 3D Cartesian basis , that vector can be expressed as

where are the time-dependent coefficients of the vector, aka the coordinates of . Here, could be the standard unit basis , for instance. The vector contains sufficient information to specify the location of point P at any time .

Velocity Vectors

The velocity vector is the time derivative of the position vector:

The vector represents the instantaneous rate of change of position and points in the direction of motion.
Assuming the same, inertial basis, we can write in component form:

Its magnitude, , is the instantaneous speed.

Heads up!

Velocities and acceleration vectors only look like this when we assume that the basis vectors do not change with time, i.e. . This is not the case in accelerated frames.

Acceleration Vectors

The acceleration vector is the time derivative of the velocity vector, or equivalently the second derivative of the position vector with respect to time:

In Cartesian component form:

Acceleration captures both changes in the magnitude and direction of the velocity vector.
For example, in curvilinear motion (common in orbital mechanics), acceleration may have both tangential and normal components.