A tracking post records the position of a small rover

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Vector calculus and field geometry · Vector calculus and field geometry

Mathematical Methods for Physics, Ch Curvilinear and orthogonal coordinates

Problem

A tracking post records the position of a small rover, relative to the post, in plane polar coordinates as r(t)=4.0 m+(0.050 m s−2)t2,θ(t)=(0.020 rad s−2)t2,r(t) = 4.0\,\mathrm m + (0.050\,\mathrm{m\,s^{-2}})t^2, \qquad \theta(t) = (0.020\,\mathrm{rad\,s^{-2}})t^2, with tt the time in seconds after the rover sets off. The rover is treated as a point that follows these formulas exactly. ((a)) Find rr, r˙\dot r, r¨\ddot r, θ˙\dot\theta and θ¨\ddot\theta at t=5.0 st=5.0\,\mathrm s. ((b)) Use the plane-polar line element ds2=dr2+r2 dθ2\mathrm ds^2=\mathrm dr^2+r^2\,\mathrm d\theta^2 to find the rover's speed at t=5.0 st=5.0\,\mathrm s. ((c)) Evaluate the four terms of r¨=(r¨−rθ˙2)r^+(rθ¨+2r˙θ˙)θ^\ddot{\mathbf r}=(\ddot r-r\dot\theta^2)\hat{\mathbf r}+(r\ddot\theta+2\dot r\dot\theta)\hat{\boldsymbol\theta} separately at t=5.0 st=5.0\,\mathrm s, name each term, and give the acceleration vector and its magnitude. ((d)) The first version of the post's software differentiated r=rr^\mathbf r = r\hat{\mathbf r} as if r^\hat{\mathbf r} were a fixed vector. What speed did it report at t=5.0 st = 5.0\,\mathrm s, which part of the velocity did it lose, and which derivative did it leave out?

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