Two bodies of masses m₁ and m₂ moving on the same direction with velociities u₁ and u₂ collide. The velocities after collision are V₁ and V₂. If each sphere loses the same amount of kinetic energy, then

Explanation

For body 1:

ΔKE₁ = ½m₁(u₁² - V₁²)

For body 2:

ΔKE₂ = 1/2m₂(u₂² - V₂²)

Since both bodies lose the same amount of kinetic energy: 
m₁(u₁² - V₁²) = m₂(u₂² - V₂²)

The total momentum before and after the collision is conserved: m₁u₁ + m₂u₂ = m₁V₁ + m₂V₂

From the energy equation:

u₁² - V₁² = (m₂/m₁)(u₂² - V₂²)

Using the identity a² - b² = (a + b)(a - b):

(u₁ + V₁)(u₁ - V₁) = (m₂/m₁)(u₂ + V₂)(u₂ - V₂)

From the momentum conservation equation, divide throughout by m₁ + m₂:

u₁ - V₁ = (m₂/m₁)(V₂ - u₂)

Substituting back into the energy equation and simplifying, we find:

u₁ + u₂ = V₁ + V₂

Final Answer

u₁ + u₂ - V₁ - V₂ = 0