Ranking final velocity after collisions

Problem

Ranking. Collisions between a $1\;\text{kg}$ ball moving at $5\;\text{m/s}$ and a ball of different mass at rest:

• (A) elastic with $m_2 = 1\;\text{kg}$;
• (B) elastic with $m_2 = 10\;\text{kg}$;
• (C) inelastic with $m_2 = 1\;\text{kg}$;
• (D) inelastic with $m_2 = 0{,}1\;\text{kg}$.

Rank the final velocity of $m_2$ from smallest to largest.

Solution

Elastic collision (stationary target): $v_2 = \dfrac{2 m_1}{m_1 + m_2}\,v$. $v_{2,A} = \frac{2\cdot 1}{2}\cdot 5 = 5 \qquad v_{2,B} = \frac{2\cdot 1}{11}\cdot 5 = 0{,}91$

Inelastic collision (stick together): $v_2 = \dfrac{m_1}{m_1 + m_2}\,v$. $v_{2,C} = \frac{1}{2}\cdot 5 = 2{,}5 \qquad v_{2,D} = \frac{1}{1{,}1}\cdot 5 = 4{,}55$

Ordering $0{,}91 < 2{,}5 < 4{,}55 < 5$: $\ev{\text{B} < \text{C} < \text{D} < \text{A}}$

Links

Topics: Momentum and collisions Concepts: Elastic collision · Inelastic collision Skills: Ranking reasoning