A rocket accelerates straight up by ejecting gas downwards . In small time interval Deltat , it ejects a gas of mass Deltamat a relative speed u . Calculate kE of the entire system at t+Deltat and tand show that the device that ejects gas does work = (1/2) Delta"mu"^(2) in this time interval (negative gravity ).

A rocket accelerates straight up by ejecting gas downwards . In small

1.	A rocket accelerates straight up by ejecting gas downwards . In small time interval Deltat , it ejects a gas of mass Deltamat a relative speed u . Calculate kE of the entire system at t+Deltat and tand show that the device that ejects gas does work = (1/2) Delta"mu"^(2) in this time interval (negative gravity ).
Answer» Solution :M =mass of rocket at t v = velocity of rocket at t `Deltam ` = mass of ejected GAS in `Deltat` u = relative SPEED of ejected gas CONSIDER at time t + `Deltat` `(KE)_(t) +(KE)_(Deltat) = `KE of rocket + Ke of `= 1/2 (M-Deltam) (v+Deltav)^(2) +1/2Deltam (v-u)^(2)` `1/2 Mv^(2) +Mv Deltav -Deltamvu +1/2 Delta"mu"^(2)` `(KE)_(t) =` KE of the rocket at time t = `1/2 Mv^(2)` `DeltaK - (KE)_(t) +(KE)_(Deltat) -(KE)_(t)` `(MDeltav - Delta"mu") v +1/2 Delta" mu"^(2)` Hene , `M (dv)/(dt) =(dc)/(dt) \|u\|` ` :. M Deltav = Delta"mu"` `:. DeltaK = 1/2 Delta"mu"^(2)` Now , by work - ENERGY theorem , `DeltaK = Delta W ` ` :. Delta W = 1/2 Delta"mu"^(2)`