Question

A rocket has initail mass M begins to move from space , with an exhaust constant speed . Find the mass of the rocket while has the maxmum momentum.

Answer 1

Answer:

$m=\frac{m_{0}}{e}$

Explanation:

Equation of the rocket is,

$m\frac{dv}{dt} =F-v'\frac{dm}{dt}$

Here, v' is the relative velocity of rocket.

In space F is zero.

So,

$m\frac{dv}{dt} =-v'\frac{dm}{dt}\\dv=-v'\frac{dm}{m} \\v=-v'ln\frac{m}{m_{0} }$

Now the momentum can be obtained by multiply by m on both sides.

$P=-v'mln\frac{m}{m_{0} }$

Now for maxima, $\frac{dP}{dm}=0$

$-v'ln\frac{m}{m_{0} }-v'm\frac{m_{0}}{m }m_{0=0$

Now,

$ln(\frac{m}{m_{0} } )=-1\\\frac{m}{m_{0} }=\frac{1}{e} \\m=\frac{m_{0}}{e}$

Therefore, the mass of the rocket while having maximum momentum is $\frac{m_{0}}{e}$