Suppose some projectile is traveling along the positive $x$ direction at a velocity $v$. While moving it is emitting particles, and thus losing mass. If the particles are being emitted in the same direction as motion, would the velocity of the original body increase or decrease ?
This is what I came up with :
$$p_i = mv$$
$$p_f = (m-dm)(v+dv)+udm$$ (Here, $u$ is the velocity of the emitted mass in the same direction)
Equating momentum on both sides, we get :
$$ mv = mv+mdv-vdm-dmdv+udm$$ $$0=mdv+(u-v-dv)dm$$
Now, if the particles are emitted in the direction of motion, let us find the relative velocity of $dm$ relative to $m$.
$v_{rel}=v_{dm \space w.r.t\space outside \space observer} + v_{outside \space observer \space w.r.t \space m} = u-(v+dv)$
Hence we can write the following: $$0=mdv+v_{rel}dm$$ Thus, $$\int_{v_I}^{v_f} dv = -v_{rel}\int_{m_i}^{m_f} \frac{dm}{m}\space\space(m_i>m_f)$$ Hence, $$\Delta v = -v_{rel}\ln\frac{m_f}{m_i} = v_{rel}\ln\frac{m_i}{m_f}$$
$$\Delta v > 0$$
This doesn't make sense to me. Shouldn't the body slow down instead of going faster now? If it emits mass in the opposite direction, then it goes faster, like a rocket. But here, I must have made some mistake. Maybe my mistake is with the limits of integration of mass? Can someone clear this doubt for me?
