How does the kinetic energy of the marble change as its potential energy changes?

• The kinetic energy of a marble changes inversely with its potential energy as it moves under the influence of gravity, assuming no external forces such as friction or air resistance. This is due to the conservation of mechanical energy, which is the sum of kinetic energy (KE) and potential energy (PE).
• When a marble is at a higher position, it has greater potential energy due to its position in a gravitational field. As the marble is released and starts to fall, its height decreases, which means its potential energy is converted into kinetic energy. This results in the marble speeding up as it falls, which increases its kinetic energy.
• Mathematically, the relationship between potential energy (PE), kinetic energy (KE), and mechanical energy (ME) can be expressed as: ME = PE + KE
• In the absence of non-conservative forces, the mechanical energy remains constant: ME_initial = ME_final
• So if the marble starts from a height h1 (with initial kinetic energy of 0 if it was at rest), the initial mechanical energy will be:

ME_initial = PE_initial + KE_initial ME_initial = m * g * h1 + 0

Where: m = mass of the marble g = acceleration due to gravity h1 = initial height of the marble

• As the marble falls to a lower height h2, its potential energy decreases, and this energy is transferred to kinetic energy:

ME_final = PE_final + KE_final m * g * h1 = m * g * h2 + 1/2 * m * v^2

• Where v is the velocity of the marble at the lower height h2.

Since ME_initial = ME_final, it is clear that as the potential energy decreases (because h2 < h1), the term 1/2 * m * v^2, which represents the kinetic energy, must increase to keep the mechanical energy constant.