Problem — Ranking kinetic energies (four objects)

Problem

Ranking. Four objects have the following masses $m$ and speeds $v$: (A) $m = 2\;\mathrm{kg}$, $v = 3\;\mathrm{m/s}$. (B) $m = 4\;\mathrm{kg}$, $v = 1\;\mathrm{m/s}$. (C) $m = 1\;\mathrm{kg}$, $v = 5\;\mathrm{m/s}$. (D) $m = 3\;\mathrm{kg}$, $v = \sqrt{6}\;\mathrm{m/s}$. Rank the kinetic energies from smallest to largest.

Solution

Compute for each object $E_k = \tfrac{1}{2}mv^2$: $E_A = \frac{1}{2}\cdot 2\cdot 3^2 = 9\;\mathrm{J}$ $E_B = \frac{1}{2}\cdot 4\cdot 1^2 = 2\;\mathrm{J}$ $E_C = \frac{1}{2}\cdot 1\cdot 5^2 = 12{,}5\;\mathrm{J}$ $E_D = \frac{1}{2}\cdot 3\cdot (\sqrt{6})^2 = \frac{1}{2}\cdot 3\cdot 6 = 9\;\mathrm{J}$ Comparison: $E_B = 2\;\mathrm{J}$, $E_A = E_D = 9\;\mathrm{J}$, $E_C = 12{,}5\;\mathrm{J}$. Note that A and D have the same kinetic energy, despite having different masses and speeds: D’s greater mass compensates for its smaller speed. $\ev{E_B < E_A = E_D < E_C}$

Links

Topics: Work and energy Concepts: Kinetic energy Skills: Ranking reasoning