A drone flying over a landing pad is tracked in the plane using the polar chart

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Tensors and differential geometry · Tensors and differential geometry

Invariance, Ch Manifolds, charts and tangent spaces

Problem

A drone flying over a landing pad is tracked in the plane using the polar chart (r,θ)(r,\theta), with x=rcos⁡θx=r\cos\theta, y=rsin⁡θy=r\sin\theta giving the Cartesian chart. At the instant the drone is at r=2 mr=2\,\mathrm{m}, θ=π/2\theta=\pi/2, its velocity in the polar chart is v=2 ∂r+3 ∂θv=2\,\partial_r+3\,\partial_\theta (with vrv^r in m s−1\mathrm{m\,s^{-1}} and vθv^\theta in rad s−1\mathrm{rad\,s^{-1}}). Find the components (vx,vy)(v^x,v^y) of vv in the Cartesian chart at that instant.

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