The correct answer is $\boxed{\frac{1}{9}}$.
The coefficient of restitution is a measure of how much energy is lost when two objects collide. A coefficient of restitution of $\frac{1}{3}$ means that the ball rebounds with $\frac{1}{3}$ of the energy it had before the collision.
The kinetic energy of an object is given by the equation $KE = \frac{1}{2}mv^2$, where $m$ is the mass of the object and $v$ is its velocity.
When the ball hits the wall, its velocity changes from $v_i$ to $v_f$. The change in kinetic energy is given by the equation $\Delta KE = \frac{1}{2}m(v_f^2 – v_i^2)$.
The coefficient of restitution is defined as the ratio of the final velocity to the initial velocity, so $e = \frac{v_f}{v_i}$.
Substituting this into the equation for the change in kinetic energy, we get $\Delta KE = \frac{1}{2}m(v_f^2 – v_i^2) = \frac{1}{2}m(v_f^2 – e^2v_i^2)$.
The fraction
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