Mechanical Energy Transfer: Calculating Potential Energy Change from Ball's Initial to Maximum Height

To determine the amount of mechanical energy transferred from the ball to the environment, we need to compare the potential energy of the ball at its initial and final heights. The potential energy (PE) of an object at a certain height is given by the formula: \[ \text{PE} = mgh \] where: - \( m \) is the mass of the object (in kilograms), - \( g \) is the acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \)), - \( h \) is the height (in meters). First, calculate the initial potential energy at the initial height of 3 meters: \[ \text{PE}_\text{initial} = mgh_\text{initial} \] \[ \text{PE}_\text{initial} = 0.08 \, \text{kg} \times 9.8 \, \text{m/s}^2 \times 3 \, \text{m} \] \[ \text{PE}_\text{initial} = 0.08 \times 9.8 \times 3 \] \[ \text{PE}_\text{initial} = 2.352 \, \text{J} \] Next, calculate the potential energy after the bounce at the final height of 1.2 meters: \[ \text{PE}_\text{final} = mgh_\text{final} \] \[ \text{PE}_\text{final} = 0.08 \, \text{kg} \times 9.8 \, \text{m/s}^2 \times 1.2 \, \text{m} \] \[ \text{PE}_\text{final} = 0.08 \times 9.8 \times 1.2 \] \[ \text{PE}_\text{final} = 0.9408 \, \text{J} \] The mechanical energy transferred to the environment is the difference between the initial and final potential energy: \[ \Delta \text{PE} = \text{PE}_\text{initial} - \text{PE}_\text{final} \] \[ \Delta \text{PE} = 2.352 \, \text{J} - 0.9408 \, \text{J} \] \[ \Delta \text{PE} = 1.4112 \, \text{J} \] The closest answer is: 1.41 J So, the correct answer is \(\boxed{1.41 \, \text{J}}\).