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

`m=(m_(0))/(e)`

Step-by-step explanation:

Equation of the rocket is,

`m(dv)/(dt) =F-v'(dm)/(dt)`

Here, v' is the relative velocity of rocket.

In space F is zero.

So,

`m(dv)/(dt) =-v'(dm)/(dt)\\dv=-v'(dm)/(m) \\v=-v'ln(m)/(m_(0) )`

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

`P=-v'mln(m)/(m_(0) )`

Now for maxima,
`(dP)/(dm)=0`

`-v'ln(m)/(m_(0) )-v'm(m_(0))/(m )m_{0=0`

Now,

`ln((m)/(m_(0) ) )=-1\\(m)/(m_(0) )=(1)/(e) \\m=(m_(0))/(e)`

Therefore, the mass of the rocket while having maximum momentum is
`(m_(0))/(e)`