I'm working on a physics problem that looks like this
For some context we have a person on his sled represented in our first term. The second term represents the velocity of a stone thrown backwards
$$ (m_p + m_s)\frac{dv_s}{dt} + m\frac{v_{st}}{dt} = 0 $$
The answer: $$ (m_p + m_s)v_s(t) + mv_0 = 0$$
where $v_0$ means the inital velocity of the throw
I cannot understand why the answer is not this:
$$ (m_p + m_s)v_s(t) + mv_{st}(t) = 0$$
i assume it if we were interested in $v_{st}(t)$ we could do this:
$$ (m_p + m_s)v_{s0} + mv_{st}(t) = 0$$