Mathematics

Problem

Two particles move along space curves given by

\mathbf r(t)=\bigl(3t,\,t^2,\,\sqrt{t}\bigr)

and

\mathbf s(t)=\bigl(\sin t,\,e^{t},\,\cos t\bigr)

for

t>0

. (a) Compute

\dfrac{d}{dt}\bigl[\mathbf r(t)\cdot\mathbf s(t)\bigr]

and simplify your result. (b) Determine all values of

t>0

at which the position vectors

\mathbf r(t)

and

\mathbf s(t)

are orthogonal. Show how your answer follows from the expression obtained in part (a). (c) Suppose the total mechanical energy

E(t)

of the system is proportional to

\mathbf r(t)\cdot\mathbf s(t)

. Using your work from parts (a) and (b), describe qualitatively when the energy is increasing, decreasing, or momentarily constant. Justify each conclusion.

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