Final answer:
The distance over which a force acts on a billiard ball to change its kinetic energy can be calculated using work-energy principles, but there seems to be an error in the provided options or the force value, since the calculated distance does not match any options.
Step-by-step explanation:
The question asks for the distance over which a force of 200 N acted to give a stationary billiard ball (mass = 0.25 kg) a speed of 2 m/s. To find this distance, we can use the work-energy principle, which states that the work done on an object is equal to its change in kinetic energy (Work = ΔKE).
The initial kinetic energy (KEi) of the stationary ball is 0 since the ball is not moving. The final kinetic energy (KEf) after the ball moves with a speed of 2 m/s is KEf = ½ × mass × speed2 = ½ × 0.25 kg × (2 m/s)2 = 0.5 J.
Since there is no initial kinetic energy, the work done on the ball is equal to the final kinetic energy: Work = 0.5 J. Work is also defined as the force times the distance over which the force acts (Work = Force × Distance), so we can calculate the distance using: Distance = Work / Force = 0.5 J / 200 N = 0.0025 m, which is not one of the options provided.
Since the question seems to have a mistake because no option matches the calculated distance, and because the provided force seems too large for the described context, please verify the details of the problem or the possible answers. In real scenarios, we would expect the distance over which a cue acts on a billiard ball to be quite small due to the brief contact during the strike.