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Kinetic energy is the energy possessed by an object due to its motion. It's an essential part of understanding how objects move and interact within a system. In the case of the hot-air balloon, the kinetic energy can be calculated using the formula:\[ KE = \frac{1}{2}mv^2 \]where:

• \( m \) is the mass of the object,
• \( v \) is the velocity of the object.

When the balloon reaches a speed of 8.00 m/s, its kinetic energy increases from zero (since it started from rest) to a new value calculated as:\( KE = \frac{1}{2} \times 500 \times (8.00)^2 \).
This increase in kinetic energy is a critical factor in analyzing how work impacts an object's movement.

Potential energy is the energy stored by an object due to its position in a gravitational field. For the hot-air balloon ascending above the Earth's surface, its potential energy increases as it rises. The formula used to calculate this energy is:\[ PE = mgh \]where:

• \( m \) is the mass,
• \( g \) is the acceleration due to gravity, roughly 9.81 m/s²,
• \( h \) is the height above the surface.

In this exercise, the potential energy change helps us determine how high the balloon rises after work is done by external forces like wind and lift. The change in potential energy, combined with kinetic energy, accounts for the total work done on the balloon, illustrating how energy shifts between kinetic and potential forms.

The principle of conservation of energy states that energy cannot be created or destroyed in an isolated system, it can only change forms. This principle ties directly into the work-energy concept, where total work done is the sum of changes in kinetic and potential energy.
For the hot-air balloon, the energy provided by the wind and lift causes changes in its mechanical energy. The work done on the balloon, \( W = 9.70 \times 10^4 \) J, transforms into both kinetic and potential energy:

• Kinetic energy increases due to its motion.
• Potential energy increases due to its elevation.

This conservation ensures that the work done manifests as these energy forms, setting the stage to calculate the balloon's height when it achieves a specific speed.

Mechanical energy is the sum of kinetic and potential energy in a system. It serves as a measure of how energy is stored and transferred within the system. In the balloon exercise, mechanical energy changes as work is performed, transitioning the balloon from rest to motion and height.
The mechanical energy equation is:\[ ME = KE + PE \]Initially, when the balloon is at rest, all mechanical energy comes from work done by external forces. As the balloon gains altitude and speed, its mechanical energy is distributed between kinetic and potential energies. The final mechanical energy consists of:

• Kinetic energy from its velocity,
• Potential energy from its height.

Understanding mechanical energy provides a comprehensive view of how energy dynamics and the work-energy principle apply to real-world scenarios, demonstrating the interconnectedness of these core concepts.