Dr. Nuha Aljuneidi

Dynamics in Motion · Mechanical Engineering

N-T vs. Polar

ACTIVITY 06

Learning goal

One motion, two coordinate descriptions

The car has one physical velocity and one physical acceleration. Normal–tangential and polar coordinates describe those same vectors using different moving bases. Change the experiment and identify what is physical, what is geometric, and what depends only on the chosen origin.

Watch for

  1. The acceleration change when the active bend radius changes.
  2. The polar components change when you move origin O.
  3. The physical vector magnitudes remain equal in both bases.
Or click and drag the orange point O directly on the canvas.

Explore Controls

t = 0%
Path regionFirst bend
Active curvatureρ = 40.0 m
Polar originCustom position
velocity v (true vector) acceleration a (true vector) ρ, e_n line (car → active bend center) r, e_r line (car → origin O — drag it!)

Normal–Tangential (e_t, e_n)

Axes ride with the car: e_t always points along its motion, e_n always points toward the road's own center of curvature C.

v_t
—
v_n
—
a_t
—
a_n
—
ρ (path curvature)
—

Polar (e_r, e_θ)

Axes are anchored to O. θ is measured the standard engineering way: counterclockwise from +x, with +y up — not raw screen pixels.

r (dist. to O)
—
θ
—
ṙ (dr/dt)
—
θ̇ (dθ/dt)
—
r̈ (d²r/dt²)
—
θ̈ (d²θ/dt²)
—

Built from those: v_r = ṙ, v_θ = rθ̇, a_r = r̈−rθ̇², a_θ = rθ̈+2ṙθ̇

v_r
—
v_θ
—
a_r
—
a_θ
—

Physical Invariant Check

Two independent computations of the same vector's magnitude — never averaged. If the tool is correct, both routes always agree.

|v| from N–T = √(v_t²+v_n²)
—
|v| from Polar = √(v_r²+v_θ²)
—
Δv = ||v|_NT − |v|_polar|
—

|a| from N–T = √(a_t²+a_n²)
—
|a| from Polar = √(a_r²+a_θ²)
—
Δa = ||a|_NT − |a|_polar|
—

Changing coordinate systems does not change the motion. v_NT = v_polar and a_NT = a_polar in magnitude, within numerical tolerance — the components differ only because the bases (e_t, e_n) vs. (e_r, e_θ) are different.

Equations (live)

Normal–Tangential

v = v e_t

a = v̇ e_t + (v²/ρ) e_n   (v̇ = 0 ⇒ a_t = 0)

—

Polar

v = ṙ e_r + rθ̇ e_θ

—

a_r = r̈ − rθ̇²

—

a_θ = rθ̈ + 2ṙθ̇

—

Interpreting the Motion

Engineering Record

Explain, in your own words, why n-t's a_n changes value partway through the curve, and why dragging the polar origin O changes r, θ, and every polar component — yet never changes the physical velocity or acceleration vectors themselves.