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Question Number 177546 by peter frank last updated on 07/Oct/22
A particle is projected vertically  upward in a constant gravitation  field with initial speed (v_0 ).show  that there is retarding proportional  to the square of the instantaneous  speed,the speed of the partical  when it returns on the initial  position is   ((v_o v_c )/( (√(v_o ^2 +v_c ^2 ))))  where v_c  is  terminal speed
$$\mathrm{A}\:\mathrm{particle}\:\mathrm{is}\:\mathrm{projected}\:\mathrm{vertically} \\ $$$$\mathrm{upward}\:\mathrm{in}\:\mathrm{a}\:\mathrm{constant}\:\mathrm{gravitation} \\ $$$$\mathrm{field}\:\mathrm{with}\:\mathrm{initial}\:\mathrm{speed}\:\left(\mathrm{v}_{\mathrm{0}} \right).\mathrm{show} \\ $$$$\mathrm{that}\:\mathrm{there}\:\mathrm{is}\:\mathrm{retarding}\:\mathrm{proportional} \\ $$$$\mathrm{to}\:\mathrm{the}\:\mathrm{square}\:\mathrm{of}\:\mathrm{the}\:\mathrm{instantaneous} \\ $$$$\mathrm{speed},\mathrm{the}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{the}\:\mathrm{partical} \\ $$$$\mathrm{when}\:\mathrm{it}\:\mathrm{returns}\:\mathrm{on}\:\mathrm{the}\:\mathrm{initial} \\ $$$$\mathrm{position}\:\mathrm{is}\:\:\:\frac{\boldsymbol{\mathrm{v}}_{\boldsymbol{\mathrm{o}}} \boldsymbol{\mathrm{v}}_{\boldsymbol{\mathrm{c}}} }{\:\sqrt{\boldsymbol{\mathrm{v}}_{\boldsymbol{\mathrm{o}}} ^{\mathrm{2}} +\boldsymbol{\mathrm{v}}_{\boldsymbol{\mathrm{c}}} ^{\mathrm{2}} }}\:\:\boldsymbol{\mathrm{where}}\:\boldsymbol{\mathrm{v}}_{\boldsymbol{\mathrm{c}}} \:\boldsymbol{\mathrm{i}}\mathrm{s} \\ $$$$\mathrm{terminal}\:\mathrm{speed} \\ $$

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