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Motion in polar coordinates
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Radial and transverse components of velocity and acceleration.
v = ṙ r̂ + rθ̇ θ̂In polar coordinates the velocity has a radial part ṙ and a transverse part rθ̇. The acceleration has radial (r̈ − rθ̇²) and transverse (rθ̈ + 2ṙθ̇) parts; the last term is the Coriolis term.
For circular motion ṙ = 0, so the radial acceleration is −rθ̇² = −v²/r.
Unit vectors r̂ and θ̂ change direction as the particle moves.
In more detail
Polar coordinates describe position by a distance r from the origin and an angle θ. Velocity then has radial and transverse components, which is useful for orbits and rotating systems.
Real-world uses
- Radar systems describe target positions in polar coordinates.
- Analysing planetary motion is simpler in polar form.
- Planetary orbits and the paths of charged particles in magnetic fields are often easiest to describe in polar coordinates.
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Frequently asked questions
When are polar coordinates useful?
When the motion has circular or radial symmetry, such as orbits or rotating arms.
What is the radial velocity?
The rate of change of the distance from the origin.
Why do polar coordinates help?
They match the geometry of circles and spirals, so the motion splits neatly into radial and angular parts.
Related topics in Kinematics
Non-uniform circular motion · Angular acceleration · Rotational kinematics equations · Motion in three dimensions · Parametric motion · Galilean transformation