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

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9 years ago

```							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.

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9 years ago
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