a ball bounces directly upon another ball at rest and it self brought to rest by the impact.if half of initial kinetic energy is destroyed in the collision ,the co-efficient of restitution is?

To find the coefficient of restitution, we need to use the concept of conservation of energy. The coefficient of restitution (e) is a measure of the elasticity of a collision and is defined as the ratio of the relative velocity of separation to the relative velocity of approach between two objects after a collision.

Let's define the variables in the problem:
v₁ = initial velocity of the moving ball
v₂ = initial velocity of the ball at rest
v₁f = final velocity of the moving ball
v₂f = final velocity of the ball at rest

According to the problem, half of the initial kinetic energy is destroyed, which means that half of the initial kinetic energy is converted into other forms of energy (like heat or deformation) during the collision. Therefore, we have:

(1/2)mv₁² - (1/2)mv₁f² = (1/2)mv₁²

This equation represents the conservation of kinetic energy before and after the collision.

Now, we can rewrite this equation as:

(1/2)mv₁² - (1/2)mv₁f² = (1/2)mv₁²(e²)

Dividing both sides of the equation by (1/2)m, we get:

v₁² - v₁f² = v₁²(e²)

Rearranging the equation, we have:

v₁f² = v₁² - v₁²(e²)

Simplifying further, we have:

v₁f² = v₁²(1 - e²)

Now, since the ball at rest doesn't move after the collision, its final velocity (v₂f) is 0.

Therefore, we have:

v₁f² + v₂f² = v₁² + v₂²

Substituting v₁f = 0 and v₂f = 0, we get:

0 + 0 = v₁² + v₂²

Simplifying, we have:

v₁² = v₁² + v₂²

Now, let's substitute v₁f² = v₁²(1 - e²), and rewrite the equation:

v₁²(1 - e²) = v₁² + v₂²

Rearranging the equation, we get:

v₁² - v₁²e² = v₁² + v₂²

Simplifying further, we have:

-v₁²e² = v₂²

Dividing both sides by v₁², we get:

-e² = (v₂/v₁)²

Finally, taking the square root of both sides, we have:

√-e² = v₂/v₁

Since v₂/v₁ is the relative velocity of separation to the relative velocity of approach, it is equal to the coefficient of restitution (e). Therefore, we have:

e = √-e²

The square root of a negative value is not meaningful in this context, so we can conclude that the coefficient of restitution, in this case, is undefined.