Three balls lie in a line on a smooth surface. The first (m₁ = 4 kg) moves toward a stationary middle ball of mass m₂, followed by a stationary third ball (m₃ = 9 kg). The collisions are elastic and head-on. What m₂ (in kg) maximizes the third ball's speed after successive collisions?
Physics · Mixed Chapter
JEE Main Practice

Question

Three balls lie in a line on a smooth surface. The first (m₁ = 4 kg) moves toward a stationary middle ball of mass m₂, followed by a stationary third ball (m₃ = 9 kg). The collisions are elastic and head-on. What m₂ (in kg) maximizes the third ball's speed after successive collisions?

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1: After the first elastic collision, speed of m₂ is v₂ = [2m₁/(m₁+m₂)]v₀. Step 2: After colliding elastically with m₃ at rest, v₃ = [2m₂/(m₂+m₃)]v₂ = 4m₁m₂v₀/[(m₁+m₂)(m₂+m₃)]. Step 3: Differentiate ln(v₃) with respect to m₂: 1/m₂ − 1/(m₁+m₂) − 1/(m₂+m₃) = 0. Step 4: This reduces to m₂² = m₁m₃ = 4×9 = 36; thus m₂ = 6 kg.