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The unit polar basis {er, eθ} is related to the standard Cartesian basis {i. j} as follows:
er = cos θi + sin θj, eθ= - sin θi + cos θj
1. Verify that
der/dt = dθ/dt · ( -sin θi + cos θj) = (dθ/dt) eθ; (deθ/dt) = -dθ/dt · (cos θi + sin θj) = -(dθ/ dt) er
2. Relative to polar coordinates, the position vector of an object moving in the plane can be written as
r (t) = r(t)er.
Verify that the velocity vector is
v(t) = (dr/dt) er + r (dθ/dt)] eθ,
and then show that the acceleration vector is
a(t) = [(d2r/dt2) - r(dθ/dt)2]er + [r(d2θ/dt2) + 2(dr/dt) · (dθ/dt)] eθ.
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