Question Number 23066 by Tinkutara last updated on 25/Oct/17
$$\mathrm{A}\:\mathrm{rocket}\:\mathrm{accelerates}\:\mathrm{straight}\:\mathrm{up}\:\mathrm{by} \\ $$$$\mathrm{ejecting}\:\mathrm{gas}\:\mathrm{downwards}.\:\mathrm{In}\:\mathrm{a}\:\mathrm{small} \\ $$$$\mathrm{time}\:\mathrm{interval}\:\Delta{t},\:\mathrm{it}\:\mathrm{ejects}\:\mathrm{a}\:\mathrm{gas}\:\mathrm{of}\:\mathrm{mass} \\ $$$$\Delta{m}\:\mathrm{at}\:\mathrm{a}\:\mathrm{relative}\:\mathrm{speed}\:{u}.\:\mathrm{Calculate}\:\mathrm{KE} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{entire}\:\mathrm{system}\:\mathrm{at}\:{t}\:+\:\Delta{t}\:\mathrm{and}\:{t}\:\mathrm{and} \\ $$$$\mathrm{show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{device}\:\mathrm{that}\:\mathrm{ejects}\:\mathrm{gas} \\ $$$$\mathrm{does}\:\mathrm{work}\:=\:\left(\frac{\mathrm{1}}{\mathrm{2}}\right)\Delta{mu}^{\mathrm{2}} \:\mathrm{in}\:\mathrm{this}\:\mathrm{time} \\ $$$$\mathrm{interval}\:\left(\mathrm{neglect}\:\mathrm{gravity}\right). \\ $$
Commented by Physics lover last updated on 26/Oct/17
$$“{Education}\:{is}\:{not}\:{learning}\:{facts} \\ $$$${but}\:{training}\:{of}\:{mind}\:{to}\:{think}'', \\ $$$${so}\:{truely}\:{said}\:{by}\:{Mr}.\:{Albert}\:{Einstien}. \\ $$$$ \\ $$
Commented by math solver last updated on 26/Oct/17
$$\mathrm{hey},\:\mathrm{rocket}\:\mathrm{repulsion}\:\mathrm{is}\:\mathrm{not}\:\mathrm{in}\:\mathrm{JEE} \\ $$$$\left.\mathrm{syllabus}\::\right) \\ $$
Commented by Physics lover last updated on 26/Oct/17
$${seriously}??????? \\ $$
Commented by math solver last updated on 26/Oct/17
$$\mathrm{yup}\:,\:\mathrm{i}\:\mathrm{am}\:\mathrm{serious}\: \\ $$$$\mathrm{you}\:\mathrm{can}\:\mathrm{check}\:\mathrm{on}\:\mathrm{net}\:\mathrm{also}. \\ $$$$\mathrm{or}\:\mathrm{else}\:\mathrm{ask}\:\mathrm{your}\:\mathrm{teachers}\:!! \\ $$
Commented by Physics lover last updated on 26/Oct/17
$${And}\:{i}\:{really}\:{believe}\:{its}\:{not}\:{only} \\ $$$${about}\:{getting}\:{good}\:{marks}\:{or}\:{rank}. \\ $$$${Its}\:{about}\:{enjoying}\:{in}\:{learning} \\ $$$${and}\:{satisfy}\:{your}\:{curiosity}. \\ $$
Commented by Physics lover last updated on 26/Oct/17
$${Alright},{Thanks}. \\ $$
Commented by Physics lover last updated on 26/Oct/17
$${Anyways},\:{we}\:{should}\:{clear}\:{our}\: \\ $$$${concept},{no}\:{matter}\:{whether}\:{its} \\ $$$${in}\:{syllabus}\:{or}\:{not}. \\ $$
Answered by ajfour last updated on 26/Oct/17
$$\:{Rocket}\:{equation} \\ $$$$\frac{{Mdv}}{{dt}}\:=\:−{u}\left(\frac{{dM}}{{dt}}\right)+{F}_{{ext}} \: \\ $$$$\Rightarrow\:\boldsymbol{{dv}}=\frac{\boldsymbol{{u}\Delta{m}}}{\boldsymbol{{M}}}\:\:\:…..\left({i}\right) \\ $$$${and}\:\:{dM}=−\Delta{m}\:\:\:\:{and}\:{F}_{{ext}} =\mathrm{0}\:{here} \\ $$$${Change}\:{in}\:{Kinetic}\:{energy}\:{of} \\ $$$${rocket}\:{and}\:{of}\:{gas}\:{between}\:{t}+\Delta{t} \\ $$$${and}\:{t}\:{is} \\ $$$$\Delta{K}={Mvdv}+\frac{\mathrm{1}}{\mathrm{2}}\left({dM}\right){v}^{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+\frac{\mathrm{1}}{\mathrm{2}}\left(\Delta{m}\right)\left({v}−{u}\right)^{\mathrm{2}} \\ $$$$\:\:\:\:\:={Mvdv}−\frac{{v}^{\mathrm{2}} \Delta{m}}{\mathrm{2}}+\frac{{v}^{\mathrm{2}} \Delta{m}}{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+\frac{\boldsymbol{{u}}^{\mathrm{2}} \boldsymbol{\Delta{m}}}{\mathrm{2}}−{uv}\Delta{m} \\ $$$$\:\:\:\boldsymbol{{W}}=\boldsymbol{\Delta{K}}=\frac{\boldsymbol{{u}}^{\mathrm{2}} \boldsymbol{\Delta{m}}}{\mathrm{2}}+\boldsymbol{{M}}{v}\left(\frac{\boldsymbol{{u}\Delta{m}}}{\boldsymbol{{M}}}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:−\boldsymbol{{uv}\Delta{m}} \\ $$$$\:\:\:\:\:\:\:\:=\frac{\boldsymbol{{u}}^{\mathrm{2}} \boldsymbol{\Delta{m}}}{\mathrm{2}}\:\:. \\ $$
Commented by Tinkutara last updated on 26/Oct/17
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{Sir}! \\ $$