Two bodies of masses  $m_1$ and $m_2$ moving on the same direction with velocities $u_1$ and $u_2$ collide. The velocities after collision are $V_1$ and $V_2$ . If each sphere loses the same amount of kinetic energy, then

To analyze the collision of two bodies of masses m1 and m2 moving in the same direction with velocities u₁ and u₂, we use the principles of conservation of momentum and kinetic energy loss. Let’s solve step by step:

Step 1: Conservation of Momentum

The total momentum before the collision equals the total momentum after the collision. Mathematically:

m₁u₁ + m₂u₂ = m₁V₁ + m₂V₂   (Equation 1)

This equation accounts for the motion of two bodies of masses m1 and m2 before and after the collision.

Step 2: Kinetic Energy Before and After Collision

The kinetic energy of the system before the collision is:

KE_initial = (1/2)m₁u₁² + (1/2)m₂u₂²

After the collision, the kinetic energy becomes:

KE_final = (1/2)m₁V₁² + (1/2)m₂V₂²

Step 3: Kinetic Energy Loss

Let the amount of kinetic energy lost by each of the two bodies of masses m1 and m2 be ΔKE. For the two masses:

For m₁: ΔKE₁ = (1/2)m₁u₁² - (1/2)m₁V₁²

For m₂: ΔKE₂ = (1/2)m₂u₂² - (1/2)m₂V₂²

Since both bodies lose the same amount of kinetic energy, we equate:

ΔKE₁ = ΔKE₂

Step 4: Setting Up the Kinetic Energy Equation

Using the above expressions, we get:

(1/2)m₁u₁² - (1/2)m₁V₁² = (1/2)m₂u₂² - (1/2)m₂V₂²

Simplify by canceling the common factor (1/2):

m₁(u₁² - V₁²) = m₂(u₂² - V₂²)

Step 5: Factorize Using the Difference of Squares

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

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

Step 6: Final Analysis

This result demonstrates that the changes in velocity (u₁ - V₁, u₂ - V₂) and the sums of velocities (u₁ + V₁, u₂ + V₂) for the two bodies of masses m1 and m2 are proportional to their respective masses.