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Kinetic energy is a form of energy that an object possesses due to its motion. Whenever an object is in motion, it has kinetic energy. It is an essential concept in physics because it relates directly to the movement of objects. The formula for kinetic energy is given by:

• \[ KE = \frac{1}{2} m v^2 \]

Here, \( m \) is the mass of the object, and \( v \) is its velocity. In the context of our helicopter exercise, the helicopter gained kinetic energy as it accelerated from rest to a speed of 7.0 m/s. This transition from a stationary position to motion builds up kinetic energy.
Calculating kinetic energy helps us understand the increase in energy as the speed of the helicopter rises. By plugging the given mass and velocity into the equation, we can find out how much energy the helicopter gained solely from its motion. This analysis is crucial for evaluating the energy transitions involved in the helicopter's ascent.

Gravitational potential energy is the energy stored in an object due to its position relative to the Earth. Any time an object is raised to a height, it gains gravitational potential energy. This energy depends on three factors:

• The mass of the object (\( m \))
• The height (\( h \)) above the ground
• The acceleration due to gravity (\( g \)), approximately \( 9.8 \, \text{m/s}^2 \)

The formula for calculating gravitational potential energy is:

• \[ GPE = mgh \]

In our helicopter scenario, as it climbed to a height of 8.2 meters, it gained gravitational potential energy. This energy represents the work done against the gravitational force to lift the helicopter to its new altitude. Gravitational potential energy is crucial for understanding how much work is involved in raising the helicopter. By calculating GPE, we ascertain the energy gain due to the helicopter's change in position.

Work done is a term that describes the amount of energy transferred by a force acting on an object as it moves through a distance. In physics, work is calculated using the equation:

• \[ \text{Work} = \text{Force} \times \text{Distance} \]

Work is particularly significant when it comes to lifting objects vertically against gravitational forces, as seen with our helicopter. In this case, the work done on the helicopter is the total kinetic energy and gravitational potential energy gained during its ascent.
In exercises like the helicopter problem, calculating work reveals the overall energy required to move the helicopter against both inertia and gravity. This comprehensive approach allows us to determine the total energy expenditure, which is essential for computing the average power generated by the lifting force. Understanding work done summarizes how different energy forms collectively enable the helicopter's motion and altitude gain.