As a ball falls under the influence of gravity, its total mechanical energy remains constant, assuming we neglect air resistance and any other forms of friction. The total mechanical energy is the sum of potential energy (PE) and kinetic energy (KE).
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Potential Energy (PE): As the ball is raised to a height \(h\), it has gravitational potential energy given by the formula: \[ PE = mgh \] where \(m\) is the mass of the ball, \(g\) is the acceleration due to gravity, and \(h\) is the height.
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Kinetic Energy (KE): As the ball falls, it accelerates and gains speed, increasing its kinetic energy, which is given by: \[ KE = \frac{1}{2}mv^2 \] where \(v\) is the velocity of the ball.
As the ball falls:
- The height \(h\) decreases, leading to a decrease in potential energy (PE).
- Simultaneously, the velocity \(v\) increases, leading to an increase in kinetic energy (KE).
The energy transformation occurs such that when the potential energy decreases, the kinetic energy increases, keeping the total mechanical energy (the sum of PE and KE) constant throughout the fall: \[ \text{Total Energy} = PE + KE \]
In summary, as the ball falls, energy is converted from potential energy to kinetic energy, but the total mechanical energy remains constant if we ignore external forces like air resistance.