What is the difference in potential energy when lifting a ball versus using an inclined plane?

When analyzing the potential energy associated with lifting a ball vertically and using an inclined plane, it's essential to focus on the concept of gravitational potential energy, which is calculated using the formula (PE = mgh), where (m) represents mass, (g) represents acceleration due to gravity, and (h) is the height the object is raised.

In both scenarios—lifting the ball straight up and moving it up an inclined plane—the height (h) ultimately determines the potential energy gained by the ball. If both the vertical lift and the incline reach the same vertical height, the change in potential energy is identical because it depends exclusively on that height, mass, and gravitational force.

Thus, the amount of energy gained by the ball in terms of potential energy will be the same in both cases, provided that the height reached is identical. Understanding this relationship is crucial, as it highlights that while the methods differ (one being a direct vertical lift and the other involving a gradual ascent), the potential energy change remains constant as long as the height achieved is the same.