Source: High school physics (Chinese)
Problem Sets:
Problem
Consider a collision between two objects with masses $m_1$ and $m_2$. Their velocity changes are defined as $\Delta v_1 = v_1' - v_1$ and $\Delta v_2 = v_2' - v_2$, where $v$ and $v'$ are the velocities before and after the collision, respectively.
- Prove that the ratio of their velocity changes is inversely proportional to the ratio of their masses: $\Delta v_1 / \Delta v_2 = -m_2 / m_1$.
- Using this result, discuss the characteristics of the velocity changes when a very light object collides with a very heavy object.
[Q1] By the law of conservation of momentum for the two-object system:
$$m_1 v_1 + m_2 v_2 = m_1 v_1' + m_2 v_2'$$Rearranging the terms:
$$m_1(v_1' - v_1) = m_2(v_2 - v_2')$$ $$m_1(v_1' - v_1) = -m_2(v_2' - v_2)$$Substituting the definitions $\Delta v_1 = v_1' - v_1$ and $\Delta v_2 = v_2' - v_2$:
$$m_1 \Delta v_1 = -m_2 \Delta v_2$$ $$\frac{\Delta v_1}{\Delta v_2} = -\frac{m_2}{m_1}$$[Q2] For a very light object ($m_1$) colliding with a very heavy object ($m_2$), we have $m_1 \ll m_2$, so the mass ratio $m_2/m_1 \gg 1$. From the relation $|\Delta v_1|/|\Delta v_2| = m_2/m_1$, it follows that $|\Delta v_1| \gg |\Delta v_2|$. The velocity change of the light object is much greater than that of the heavy object. The heavy object's velocity is nearly unchanged.