91Ó°ÊÓ

Kinetic energy is an essential concept in physics, describing the energy an object possesses due to its motion. It's given by the formula: \[ KE = \frac{1}{2} m v^2 \] where \( m \) is the mass of the object and \( v \) is its velocity. In the context of our helicopter exercise, the kinetic energy is calculated when the helicopter, originally at rest, starts moving upwards. Because it starts from rest, its initial kinetic energy is zero. As it reaches a velocity of \( 7.0 \ \text{m/s} \), the calculation for the kinetic energy change becomes \[ \Delta KE = \frac{1}{2} \times 810 \times (7.0)^2 = 19845 \ \text{Joules}. \]This change indicates the energy needed to accelerate the helicopter to its given speed over its ascent from rest.

Gravitational potential energy (GPE) refers to the energy stored in an object due to its height above the ground. This energy is dependent on three variables: the object's mass, the height from which it's elevated, and the gravitational acceleration. It's given by the expression:\[ \Delta U = mgh \]For our helicopter, the mass \( m \) is \( 810 \ \text{kg} \), \( g \) represents the acceleration due to gravity at \( 9.8 \ \text{m/s}^2 \), and \( h \) is the height achieved, specifically \( 8.2 \ \text{m} \). Plugging these values into the equation, we calculate\[ \Delta U = 810 \times 9.8 \times 8.2 = 65285.6 \ \text{Joules}. \]This result symbolizes the energy gained by the helicopter because of its lifted position above the hospital roof.

The Work-Energy Principle is a fundamental concept that connects the total work done on an object with its change in kinetic and potential energy. According to this principle, when work is performed on an object, it is transformed into either kinetic or potential energy or remains as work done against friction. In simpler terms, work done equals the total change in mechanical energy:\[ W_{total} = \Delta KE + \Delta U \]Applied to the helicopter scenario, the total work done from lifting it combines both kinetic and gravitational energy changes. Calculation yields:\[ W_{total} = 19845 + 65285.6 = 85130.6 \ \text{Joules}. \]This sum represents the complete amount of energy exerted by the lifting forces as the helicopter ascends.

Unit conversion is a practical tool in scientific calculations, allowing us to express measurements in different units that may be more meaningful or practical for the context. For instance, power expressed in watts can also be converted to horsepower for certain applications.The formula connecting watts and horsepower is:\[ 1 \ \text{horsepower} = 746 \ \text{Watts}. \]In our exercise, the calculated average power in watts is \[ \frac{85130.6}{3.5} = 24322.8 \ \text{Watts}. \]To transform this into horsepower:\[ \frac{24322.8}{746} \approx 32.6 \ \text{horsepower}. \]Converting units not only helps in understanding various aspects of power rating but also facilitates communication in industries where different units are customarily used.