Question

A spaceship of mass m has velocity v in the positive x direction of an inertial reference frame. A mass dm is fired out the rear of the ship with constant exhaust velocity (-v0) with respect to the spaceship. a) using conservation of momentum, show that dv/v0 = dm/m, b) By integration, find the dependence of v on m if v1 and m1 are the initial values. c) Can the acceleration be constant if dm/dt, the burning rate is constant.

Answer 1

Given,

Mass of the spaceship = m

Velocity of the spaceship = v

So, initial momentum of the spaceship (p) = mv

As per question, $dm$ mass fired with $v_o$ velocity in opposite direction.

Applying the conservation of momentum,

Initial momentum = final momentum

$p=mv$

$\Rightarrow \frac{dp}{dt}=\frac{dm}{dt}v_o+m\frac{dv}{dt}$

change in momentum will remain unchanged,

so, $\dfrac{dp}{dt}=0$

$\Rightarrow \dfrac{dm}{m}=-\frac{dv}{v_o}$

$d\overrightarrow{J}=d\overrightarrow{p}$

$\Rightarrow \overrightarrow{p_f}-\overrightarrow{p_i}=-mgdt\hat{j}$

$\Rightarrow mdv-dm_gu=-mgdt$

$\Rightarrow mdv=-dm u-mgdt$

$\Rightarrow dv=-u\frac{dm}{m}-gdt$