Physics problem. Already asked this question yesterday but no one replied so I\'
ID: 1503635 • Letter: P
Question
Physics problem. Already asked this question yesterday but no one replied so I'm trying again.
Thank you
We derived the rocket equation in class for a rocket in empty space with no gravity. Re-derive the equation for a rocket launching from Earth, with the force of gravity (assume it's constant, i.e., the rocket never gets very far from the Earth's surface) acting on it. What is the final velocity of the rocket after it has burned all its fuel? Given the results, why do rocket launches looks so spectacular?Explanation / Answer
Let M be the mass of rocket including fuel and V its velocity at any instant in laboratory frame of reference
let in a time interval dt , an amount of mass dM is ejected from the rocket in the form of gas jet
suppose -u is the velocity of gas jet w.r.to rocket .then the velocity of gas jet in the laboratory frame will be v-u
Let v-u =Vrel
the rate of chaange of momentum of jet issuing out of the rocket i.e., force acting on the jot
= rate of change of mass of rocket due to escape of hot gases* velocity
=dM/dt *(v-u)
from the newton's third law of motion this is equal to the thrust on the rocket tending to move it in the forward direction
thrust acting on the rocket to move it forward
T= dM/dt (v-u).....................................................(1)
Let F ext be the external force acting on the rocket
here F ext =-Mg where Mg=the weight of the rocket
Then
Force acting on the rocket in forward direction
F resultant = dM/dt (v-u) - Mg.............................................(2)
from newton's second law the force on the rocket
= d/dt (Mv).................................(3)
from eq 2 and eq 3 we get
d/dt (Mv) =dM/dt(v-u) -- Mg............................................(4)
M dv/dt +v dM/dt = v dM/dt -- u dM/dt - Mg
M dv/dt = - u dM/dt --Mg .....................................(5)
this is first rocket equation
now dv/dt = - u 1/M dM/dt - g
consider the magnitude of v u and g we have
dv = - u dM/dt - g dt.................................................(6)
now we applying intergating on both side with limits V and Vo M and Mo , T and To
we get
V- Vo = -u Log{M/Mo} - gt
V=Vo + u Log {Mo/M} - gt...............................(7)
this is the equation of rocket burned all fuel
special cases:-
(1) if the force of gravity is ignored then
V=Vo +u Log{Mo/m}
(2) further if the initial velocity of rocket is zero
V= u Log {Mo/M}