Ranking kinetic energies

Problem

Same objects as the previous problem: (A) car $1500\;\text{kg}$ at $20\;\text{m/s}$; (B) ball $0{,}1\;\text{kg}$ at $30\;\text{m/s}$; (C) bicycle plus cyclist $80\;\text{kg}$ at $10\;\text{m/s}$; (D) golf ball $0{,}05\;\text{kg}$ at $80\;\text{m/s}$. Rank the kinetic energies from smallest to largest and comment on whether the order is the same as for $p$.

Solution

Kinetic energy is $E_\text{kin} = \tfrac{1}{2}\,m\,v^2$. Computing for each object: $\begin{aligned} E_A &= \tfrac{1}{2}\cdot 1500 \cdot 20^2 = 3{,}0\cdot 10^5\ \mathrm{J}\\ E_B &= \tfrac{1}{2}\cdot 0{,}1 \cdot 30^2 = 45\ \mathrm{J}\\ E_C &= \tfrac{1}{2}\cdot 80 \cdot 10^2 = 4\,000\ \mathrm{J}\\ E_D &= \tfrac{1}{2}\cdot 0{,}05 \cdot 80^2 = 160\ \mathrm{J} \end{aligned}$ Order from smallest to largest: $E_B < E_D < E_C < E_A$. In this case the order coincides with that of momentum ($p_B < p_D < p_C < p_A$), but this is a coincidence tied to these particular numbers: $p = mv$ grows linearly with speed, while $E = \tfrac{1}{2}mv^2$ grows with the square. A light, very fast object can therefore have small $p$ but relatively large $E$: in general the two criteria give different orderings. $\ev{E_B < E_D < E_C < E_A \qquad (45 < 160 < 4\,000 < 3\cdot 10^5\ \mathrm{J})}$

Links

Topics: Momentum and collisions Concepts: Kinetic energy · Momentum Skills: Ranking reasoning