< b="" />Problem Statement:< />
A particle moving with velocity 'u' collides with another identical particle at rest. The coefficient of restitution between them is 'e'. We need to find their velocities after the collision.

< b="" />Solution:< />

< b="" />Step 1: Understanding the Coefficient of Restitution:< />
The coefficient of restitution, denoted by 'e', is a measure of the elasticity of a collision between two objects. It determines how much kinetic energy is conserved during the collision. The value of 'e' lies between 0 and 1, where:
- e = 1 represents a perfectly elastic collision, where kinetic energy is conserved.
- e = 0 represents a perfectly inelastic collision, where kinetic energy is completely lost.

< b="" />Step 2: Analyzing the Collision:< />
In this problem, we have two identical particles colliding. One particle is moving with velocity 'u' (initial velocity) and the other particle is at rest. Let's assume that after the collision, the first particle has a final velocity 'v1' and the second particle has a final velocity 'v2'.

< b="" />Step 3: Applying the Coefficient of Restitution Equation:< />
According to the coefficient of restitution equation, the relative velocity of separation (v1 - v2) after the collision is equal to the negative of the relative velocity of approach (u - 0) before the collision, multiplied by the coefficient of restitution (e).

So, we have:
v1 - v2 = -e * (u - 0)
v1 - v2 = -e * u

< b="" />Step 4: Conservation of Momentum:< />
In an isolated system, the total momentum before the collision is equal to the total momentum after the collision. Since the particles are identical, their masses are also the same. Let's assume the mass of each particle is 'm'.

Before the collision:
Total momentum = m * u + m * 0 = m * u

After the collision:
Total momentum = m * v1 + m * v2 = m * (v1 + v2)

Since the total momentum is conserved, we can write:
m * u = m * (v1 + v2)

Simplifying the equation, we get:
u = v1 + v2
v2 = u - v1

< b="" />Step 5: Solving the Equations:< />
Now, we have two equations:
v1 - v2 = -e * u
u = v1 + v2

We can substitute the value of v2 from the second equation into the first equation:
v1 - (u - v1) = -e * u
2 * v1 - u = -e * u
2 * v1 = (1 - e) * u
v1 = (1 - e) * u / 2

Substituting the value of v1 into the equation u = v1 + v2, we get:
u = (1 - e) * u / 2 + v2
v2 = u - (1 - e) * u / 2
v2 = (2 - 1 + e) * u / 2
v2 = (1 + e) * u / 2

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