Explanation:

When the ball hits the rough vertical wall, it experiences a collision. The coefficient of restitution between the ball and the wall is given as 1/3. This coefficient of restitution determines the ratio of the final relative velocity to the initial relative velocity after the collision.

To find the fraction of kinetic energy lost by the ball, we need to compare the kinetic energy before and after the collision.

Initial Situation:
- The ball is moving on a smooth horizontal table, which means there is no loss of kinetic energy due to friction.
- Let the initial velocity of the ball be 'v' and its mass be 'm'.
- The initial kinetic energy of the ball is given by KE_initial = (1/2)mv^2.

Collision with the Wall:
- When the ball hits the rough vertical wall, it experiences a collision.
- The ball rebounds at the same angle, which means the direction of its velocity changes but the magnitude remains the same.
- Let the final velocity of the ball after the collision be 'v_f'.
- According to the coefficient of restitution, the relative velocity after the collision is given by (1/3)v.
- Therefore, the final relative velocity of the ball is v + (1/3)v = (4/3)v.
- Since the ball rebounds at the same angle, the horizontal component of the velocity remains the same while the vertical component changes direction.

Final Situation:
- After the collision, the ball continues to move on the horizontal table.
- The final kinetic energy of the ball is given by KE_final = (1/2)m(v_f)^2.

Fraction of Kinetic Energy Lost:
- The fraction of kinetic energy lost by the ball is given by (KE_initial - KE_final) / KE_initial.
- Substituting the values, we have:
Fraction of kinetic energy lost = [(1/2)mv^2 - (1/2)m((4/3)v)^2] / [(1/2)mv^2]
= [(1/2)mv^2 - (1/2)m(16/9)v^2] / [(1/2)mv^2]
= [(1/2)mv^2 - (8/9)mv^2] / [(1/2)mv^2]
= [(9/18)mv^2 - (8/9)mv^2] / [(1/2)mv^2]
= [(9/18) - (8/9)]
= (1/18)
≈ 0.056

Therefore, the fraction of kinetic energy lost by the ball is approximately 0.056 or 5.6%.