PLZ DERIVE EXPRESSION FOR ROCKET IN HCV I AM NOT UNABLE TO GET IT PLZZZZ ANY OTHER BIT MORE EXPALANATORY

Instantaneous velocity of the rocket. Consider a flight of a rocket directed vertically upwards at time t=0, the mass of the rocket + fuel = m0 and its initial velocity is v0 .

Let v be the upward velocity of the rocket at this time . Momentum at time t=mv

mass of rocket and unburnt fuel will reduce to (m-delta m) at the end of the interval delta t

let the velocity increase to (v + delta v).

momentum of rocket and unburnt fuel at time (t + delta t) = (m - delta m)( v + delta v)

total momentum at time (t + delta t) = (m - delta m)(v + delta v) + m( -v gravitation)

v gravitation is the velocity of the echaust gases with respect to the earth. It is taken as negative becasue gases move in opposite directions. to that of the rocket.

mv = (m - delta m) ( v + delta v) + m ( -v gravitation)

mv = mv + m delta v - v delta m - delta m delta v - v gravitation delta m.

m delta v = delta m ( v + v gravitation) + delta m delta v

the term delta m delta v can be dropped because of its small vale.

m delta v = delta m ( v + v gravitation)

if vr is the resultant velocity of the gases relative to the rocket. then

vr = v gravitation + (-v) = v gravitation - v

vr = -v gravitation - v

negative sign shows that vr is in a downward direction whereas the motion of the rocket is in upward direction.

m delta v = - vr delta m

delta v = - vr (delta m / m)

dv = -vr (dm/m)

integrating we get

v-v0 = -vr [ ln m -ln m0] = -vr ln [m/m0]

v = v0 + vr ln [ m0/m ] ------------------------------------------------------------- (1)

this gives velocity of rocket at any time t when the mass of the unburnt fuel is m

initial velocity v0 = 0 then

v = vr ln [m0/m] ------------------------------------------------------------------------(2)

accelaration of the rocket

differenciation of velocity gives accelaration.

accelaration is inversely propertional to m

Thrust on rocket

F = -vr dm/dt --------------------------------------------------------------------------(3)

Burn out speed.

vb = burn out speed.

vb = v0 + vr log [m0/mc]                  ------ (4) where mc is the mass of the container containing fuel.

PLEASE APPROVE

Approved