A particle of mass moving in one dimension with velocity

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Calculus of change and accumulation · Calculus of change and accumulation

Conditions of Validity, Ch Differentiability, the implicit function theorem and the licence to change variables

Problem

A particle of mass m=0.50 kgm = 0.50\ \text{kg} moving in one dimension with velocity vv, of either sign, has kinetic energy T(v)=12mv2T(v) = \tfrac12 m v^2. ((a)) Show T′′(v)>0T''(v)>0 for all vv, which makes TT strictly convex, and show that pv−T(v)pv-T(v) has a finite maximum over vv for every real pp, so that the transform is defined for every pp. ((b)) Compute L[T](p)=sup⁡v(pv−T(v))\mathcal{L}[T](p) = \sup_v\big(pv - T(v)\big), and identify the new independent variable pp physically. ((c)) Evaluate L[T](p)\mathcal{L}[T](p) at p=2.0 kg m s−1p = 2.0\ \text{kg m s}^{-1}.

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