A spaceship of mass `m_0` moves in the absence of external forces with a constant velocity `v_0`. To change the motion direction, a jet engine is switched on. It starts ejecting a gas jet with velocity u which is constant relative to the spaceship and directed at right angles to the spaceship motion. The engine is shut down when the mass of the spaceship decreases to m. Through what angle `alpha` did the motion direction of the spaceship deviate due to the jet engine operation?

Explanation

As `vecF=0`, from the equation of dynamics of a body with variable mass,
`m(dvecv)/(dt)=vecu(dm)/(dt)` or, `dvecv=vecu(dm)/(m)` (1)
Now `dvecvuarrdarrvecu` and since `vecu_|_vecv`, we must have `|dvecv|=v_0dalpha` (because `v_0` is constant)
where `dalpha` is the angle by which the spaceship turns in time `dt`.
So, `-u(dm)/(m)=v_0dalpha` or, `dalpha=-(u)/(v_0)(dm)/(m)`
or, `alpha=-(u)/(v_0)underset(m_0)overset(m)int(dm)/(m)=u/v_(0) 1n(m_0/m)`