Polar / Path Normal–Tangential Acceleration

a_{t} = v ˙ = r α, a_{n} = \frac{v ^{2}}{ρ}, a_{r} = r ¨ - r θ ˙^{2}, a_{θ} = r θ ¨ + 2 r ˙ θ ˙

Use n–t when the path is known (ρ = radius of curvature). Use r–θ when motion is given in polar form. The 2ṙθ term is Coriolis in the polar basis.

Step-by-step solved example

Car v=20 m/s, ρ=100 m, speeding up at 2 m/s². |a|?

a_n = v²/ρ toward the center of the path.

Answer: |a| = 4.47 m/s²

0/10 correct

1.If speed is constant, a_t is

2.Coriolis term 2ṙθ̇ is in

3.a_n direction is

4.v=10, ρ=25, a_n =

5.Straight path means

6.ṙ=0 (circular polar) reduces a_r to

7.Units of α are

8.Total |a|=√(a_t²+a_n²) assumes

9.θ̈ is

10.A particle in a rotating tube (ṙ≠0) feels Coriolis in the tube’s frame as