Two bodies A and B have masses in the ratio 5:1 and their kinetic energies are in the ratio 125:9. Find the ratio of their velocities
Topic: Kinetic energy
Answer
Given: m_A : m_B = 5 : 1 and KE_A : KE_B = 125 : 9
Kinetic energy, KE = ½ m v²
Taking the ratio of the two bodies: KE_A ÷ KE_B = (½ m_A v_A²) ÷ (½ m_B v_B²) = (m_A ÷ m_B) × (v_A² ÷ v_B²)
125 ÷ 9 = (5 ÷ 1) × (v_A² ÷ v_B²)
v_A² ÷ v_B² = (125 ÷ 9) ÷ 5 = 125 ÷ 45 = 25 ÷ 9
Taking the square root of both sides: v_A ÷ v_B = 5 ÷ 3
Ratio of their velocities, v_A : v_B = 5 : 3
With ratios you never need the actual masses or speeds. Write the formula, divide one body by the other, and the ½ cancels immediately.
The step people miss is the square root at the end. 25 : 9 is the ratio of the SQUARES of the velocities; the velocities themselves are in the ratio 5 : 3. Stopping at 25 : 9 is the standard wrong answer.
Divide by the mass ratio, do not multiply — the mass is already accounting for part of the energy difference, so it comes off the energy ratio to leave the velocity part behind.