Finding the masses from the CM

Problem

Inverse problem. Two masses move in opposite directions with $|v_1| = |v_2| = 5\;\text{m/s}$. Their centre of mass moves at $v_\text{cm} = +1\;\text{m/s}$. Knowing that the total mass is $10\;\text{kg}$, find $m_1$ and $m_2$.

Solution

Let $m_1$ be at $+5\;\text{m/s}$ and $m_2$ at $-5\;\text{m/s}$. The CM velocity is $v_\text{cm} = \frac{5\,m_1 - 5\,m_2}{m_1 + m_2} = \frac{5(m_1 - m_2)}{10} = 1 \;\Rightarrow\; m_1 - m_2 = 2$ With the constraint $m_1 + m_2 = 10$ we get the system: $\ev{m_1 = 6\;\text{kg}, \quad m_2 = 4\;\text{kg}}$

Links

Topics: Centre of mass Concepts: Centre of mass · Momentum Skills: Symbolic setup