Relative to polar coordinates the position vector of an

Question

The unit polar basis {e_r, e_θ} is related to the standard Cartesian basis {i. j} as follows:

e_r = cos θi + sin θj, e_θ= - sin θi + cos θj

1. Verify that

de_r/dt = dθ/dt · ( -sin θi + cos θj) = (dθ/dt) e_θ; (de_θ/dt) = -dθ/dt · (cos θi + sin θj) = -(dθ/ dt) e_r

2. Relative to polar coordinates, the position vector of an object moving in the plane can be written as

r (t) = r(t)e_r.

Verify that the velocity vector is

v(t) = (dr/dt) e_r + r (dθ/dt)] e_θ,

and then show that the acceleration vector is

a(t) = [(d²r/dt²) - r(dθ/dt)²]e_r + [r(d²θ/dt²) + 2(dr/dt) · (dθ/dt)] e_θ.