Question

Show that, if a spacecraft of total mass m(t) propels itself by ejecting exhaust

gas from its rocket motor with a relative velocity u, then its velocity v(t) satisfies:

m dv = -u dm (where v and u are vectors)

and hence, making clear any assumptions in your derivation, that the initial and

final speeds v_i and v_f are related to the initial and final masses m_i and m_f by:

v_f = v_i + u ln (m_i /m_f)

Answer 1

suppose at time t=0 the mass of rocket+fuel is and  on being ignited velocity with respect to earth is

according to conservation of linear momentum

linear momentum of mass m of the rocket at time t=vector sum of the momentum of mass (m-dm) of the rocket+mass dm moving in the form of burnt gases

ie  

..........(1)

product of two very small quantities so can be neglected

so eq (1) becomes

if vector u will be the velocity of the burnt gases relative to rocket,then

or   

(-ve sign shows that u is in the downward direction whereas the motion of the rocket (v) is in upward direction.

after substituting value of

now lets integrate this equation limit of v is from to while limit for m will be from to .

so final eq that we we after integration will be

or

here m means