Solution

Statement

$2.2.10.$ A spaceship had a velocity $v$ before the separation of the last stage of the launch vehicle. After discarding the last stage, its velocity became $1.01v$, while the separated stage moves away relative to the ship with a velocity of $0.04v$. What is the mass of the last stage if the mass of the ship is $m_0$?

Solution

Let's write the law of conservation of momentum in projection onto the axis of the ship's motion:
$$(m_0 + m_2)v = 1.01m_0 v + (1.01 - 0.04)m_2 v$$

From which we obtain:
$$0.03m_2 = 0.01m_0 \implies m_2 = m_0 / 3$$