Introduction:
The coefficient of restitution is a measure of the elasticity of a collision between two objects. It determines how much kinetic energy is conserved during the collision. In this case, we are given the initial and final heights of a ball dropped vertically onto a floor, and we need to calculate the coefficient of restitution.

Given:
- Mass of the ball (m) = 1 kg
- Initial height (h1) = 25 cm
- Final height (h2) = 9 cm

Calculating the coefficient of restitution:
The coefficient of restitution (e) can be calculated using the formula:

e = sqrt(h2 / h1)

Substituting the given values:
e = sqrt(9 cm / 25 cm)

Simplifying the equation:
e = sqrt(0.36)

Calculating the square root:
e ≈ 0.6

Therefore, the coefficient of restitution for the collision is approximately 0.6.

Explanation:
The coefficient of restitution represents the ratio of the final velocity of separation to the initial velocity of approach during a collision. In this case, we are given the initial and final heights of the ball dropped onto the floor.

When the ball is dropped from a height of 25 cm, it gains potential energy. This potential energy is converted into kinetic energy as the ball falls. When the ball collides with the floor, some of the kinetic energy is converted into potential energy as the ball rebounds.

The coefficient of restitution can be determined by comparing the initial and final heights. By using the formula e = sqrt(h2 / h1), we can calculate the coefficient of restitution.

In this case, the ball rebounds to a height of 9 cm. Plugging this value into the formula, we find that the coefficient of restitution is approximately 0.6.

This means that after the collision, the ball retains approximately 60% of its initial velocity. The rest of the energy is lost as heat, sound, and deformation of the ball and floor.

Conclusion:
The coefficient of restitution for the collision between the ball and the floor is approximately 0.6. This value indicates that the collision is somewhat elastic, as the ball rebounds to a significant height compared to its initial height.