Calculator · Engineering Mechanics
Curvilinear Motion Calculator
Planar curvilinear motion of a particle, in two explicit coordinate systems. Normal–Tangential coordinates follow the path itself; Polar coordinates measure the same motion from a fixed pole. The equations are never mixed between the two.
Coordinate system
Normal–Tangential Coordinates
Signed along ê_t: positive speeds the particle up along the path.
Results
| Relation used | a_n = v²/ρ and a = a_t ê_t + a_n ê_n |
|---|---|
| Speed v | 10 m/s |
| Tangential acceleration a_t | 3 m/s² |
| Normal acceleration a_n | 5 m/s² — towards the centre of curvature |
| Total acceleration |a| | 5.831 m/s² |
| Radius of curvature ρ | 20 m |
| Direction of a relative to the tangent | 59.036° from ê_t towards ê_n |
Path components
Schematic: ê_t is drawn along the path and ê_n towards the centre of curvature. Arrow lengths are graphically scaled; the magnitudes are the table above.
Technical reference
n–t: a_t = dv/dt, a_n = v²/ρ (ρ → ∞ ⇒ a_n = 0)
polar: a_r = r̈ − r·θ̇², a_θ = r·θ̈ + 2·ṙ·θ̇
- v, a_t, a_n
- path speed, tangential and normal (inward) acceleration[m/s, m/s²]
- ρ
- radius of curvature, ρ > 0[m]
- r, ṙ, r̈
- polar radius and its time derivatives[m, m/s, m/s²]
- θ̇, θ̈
- angular rate and angular acceleration[rad/s, rad/s²]
a_n is always reported as a non-negative magnitude towards the centre of curvature. In the polar form, 2·ṙ·θ̇ is the Coriolis-like term inside a_θ; this is planar polar kinematics of one particle, not a general rotating-frame model.
Assumptions
- · A single point particle in a plane, at one instant.
- · Normal–Tangential: ê_n points towards the local centre of curvature, so a_n is a magnitude and is never negative. Which side of the path curves is not encoded in its sign.
- · Normal–Tangential: ρ > 0, except in the explicit straight-line limit where a_n = 0.
- · Polar: the pole is fixed and θ is measured counter-clockwise from +x. The 2 ṙ θ̇ term is the Coriolis-like term of planar polar kinematics — this is not a general rotating-frame model.
Continue
A general n–t or polar state is not a constant Cartesian acceleration, which is what the Particle Motion Lab models, so no state is carried across. These tools cover the neighbouring cases exactly.