The coefficient of restitution (e) is a measure of how elastic a collision is, and it is defined as the ratio of the final relative velocity to the initial relative velocity. In this case, we have a ball of mass m moving at speed v colliding head-on with an identical ball at rest.

Let's denote the final velocities of the two balls as v1 and v2 respectively.

The conservation of momentum gives us:

m*v = m*v1 + m*v2 ----(1)

The given condition that the final kinetic energy is two thirds of the initial kinetic energy gives us:

(1/2)*m*v1^2 + (1/2)*m*v2^2 = (2/3)*(1/2)*m*v^2 ----(2)

From equation (1), we can express v2 as v - v1 and substitute this into equation (2). After simplifying, we get:

v1 = v/3

Substitute v1 into equation (1) to get v2 = 2v/3.

The coefficient of restitution e is defined as:

e = (v2 - v1) / v

Substitute v1 and v2 into the equation to get:

e = (2v/3 - v/3) / v = 1/3

So, the coefficient of restitution is 1/3.

Question

The coefficient of restitution (e) is a measure of how elastic a collision is, and it is defined as the ratio of the final relative velocity to the initial relative velocity. In this case, we have a ball of mass m moving at speed v colliding head-on with an identical ball at rest.

Let's denote the final velocities of the two balls as v1 and v2 respectively.

The conservation of momentum gives us:

m*v = m*v1 + m*v2  ----(1)

The given condition that the final kinetic energy is two thirds of the initial kinetic energy gives us:

(1/2)*m*v1^2 + (1/2)*m*v2^2 = (2/3)*(1/2)*m*v^2  ----(2)

From equation (1), we can express v2 as v - v1 and substitute this into equation (2). After simplifying, we get:

v1 = v/3

Substitute v1 into equation (1) to get v2 = 2v/3.

The coefficient of restitution e is defined as:

e = (v2 - v1) / v

Substitute v1 and v2 into the equation to get:

e = (2v/3 - v/3) / v = 1/3

So, the coefficient of restitution is 1/3.

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