Question Number 64688 by peter frank last updated on 20/Jul/19
Answered by mr W last updated on 20/Jul/19
$${M}_{\mathrm{0}} ={original}\:{mass} \\ $$$$−{u}\frac{{dm}}{{dt}}={m}\frac{{dv}}{{dt}} \\ $$$$−{u}\int_{{M}_{\mathrm{0}} } ^{\:{m}} \frac{{dm}}{{m}}=\int_{{v}_{\mathrm{0}} } ^{\:{v}} {dv} \\ $$$$−{u}\left(\mathrm{ln}\:{m}−\mathrm{ln}\:{M}_{\mathrm{0}} \right)={v}−{v}_{\mathrm{0}} \\ $$$$\Rightarrow{v}={v}_{\mathrm{0}} −{u}\:\mathrm{ln}\:\frac{{m}}{{M}_{\mathrm{0}} } \\ $$$${with}\:{m}=\frac{{M}_{\mathrm{0}} }{\mathrm{2}} \\ $$$$\Rightarrow{v}=\mathrm{20}−\mathrm{5}\:\mathrm{ln}\:\frac{\mathrm{1}}{\mathrm{2}}=\mathrm{23}.\mathrm{47}\:{m}/{s} \\ $$
Commented by peter frank last updated on 20/Jul/19
$${thank}\:{you} \\ $$