A rocket with an initial mass `m_(0)` is going up with a constant acceleration a by exhausting gases with a velocity v relative to the rocket motion, then the mass of the rocket at any instant of time is (assume that no other forces act on it)

Options

A. `m=m_(0)e^(-(at)/v)`
B. `m=m_(0)e^(-(2at)/v)`
C. `m=m_(0)e^(-(at)/(2v))`
D. `m=m_(0)e^(-(a^(2)t^(2))/(v^(2)))`

Answer

A

Explanation

For a rocket,
`m(dv)/(dt)=F_("ext")+v(dm)/(dt)`

Without any external force `(F_("ext")=0)`,
`m(dv)/(dt)=v(dm)/(dt)rArrint(dv)/v=int(dm)/m`
`rArrlogv=logm+C`
When, `v=0,m=m_(0)`
`thereforeC=-logm_(0)`
So, we have
`(Deltav)/v="log"m/m_(0)*dtorm=m_(0)e^(-at//v)`