Question

An \(1865-\mathrm{kg}\) airplane starts at rest on an airport runway at sea level. What is the change in mechanical energy of the airplane if it climbs to a cruising altitude of \(2420 \mathrm{~m}\) and maintains a constant speed of \(96.5 \mathrm{~m} / \mathrm{s}\) ?

Short answer

The change in mechanical energy of the airplane is approximately \( 52.9 \times 10^6 \text{ Joules} \).

Step-by-step solution

Calculate the Gravitational Potential Energy

The change in gravitational potential energy (GPE) when the airplane rises to a height \( h \) is given by the formula \( \Delta PE = mgh \), where \( m \) is mass, \( g \) is acceleration due to gravity \( (9.81\, \text{m/s}^2) \), and \( h \) is height. Substituting the given values, we have: \( \Delta PE = 1865\, \text{kg} \times 9.81\, \text{m/s}^2 \times 2420\, \text{m} \). Calculate \( \Delta PE \).

Substitute and Solve for Gravitational Potential Energy

Calculate \( \Delta PE \) using the equation: \( \Delta PE = 1865 \times 9.81 \times 2420 \). This results in \( \Delta PE = 44,244,693 \text{ Joules} \).

Calculate the Kinetic Energy

The kinetic energy (KE) can be calculated using the formula \( KE = \frac{1}{2}mv^2 \), where \( m \) is mass and \( v \) is velocity. Using the given constant speed of the airplane, \( KE = \frac{1}{2} \times 1865 \times (96.5)^2 \). Calculate \( KE \).

Substitute and Solve for Kinetic Energy

Substitute the values in the equation to find \( KE: KE = \frac{1}{2} \times 1865 \times 96.5^2 = 8,679,871.125 \text{ Joules} \).

Calculate the Total Change in Mechanical Energy

The total change in mechanical energy is the sum of the change in potential energy and the change in kinetic energy. That is, \( \Delta E_{total} = \Delta PE + KE \). Substitute the values: \( \Delta E_{total} = 44,244,693 + 8,679,871.125 \). Calculate \( \Delta E_{total} \).

Final Solution for Total Change in Mechanical Energy

Calculate the sum \( \Delta E_{total} = 44,244,693 + 8,679,871.125 = 52,924,564.125 \text{ Joules} \). Therefore, the change in mechanical energy of the airplane is approximately \( 52.9 \times 10^6 \text{ Joules} \).

Key concepts

Gravitational Potential Energy

Gravitational potential energy (GPE) is a type of energy an object possesses because of its position relative to Earth. It's like having a reserve of energy stored due to gravity. When you lift an object, you're storing energy in it. The higher you lift it, the more energy gets stored. This energy depends on three main factors:

• Mass (\( m \))
• Height (\( h \))
• Gravitational acceleration (\( g \) which is typically \( 9.81 \) m/s²)

You can calculate it using the formula:\[\Delta PE = mgh\]In our example, the change in gravitational potential energy is found by multiplying the mass of the airplane (1865 kg), the gravitational acceleration (\( 9.81 \) m/s²), and the height (\( 2420 \) m). This results in an energy change of 44,244,693 Joules. Imagine this energy as the amount required to lift the airplane to its cruising height without accounting for its movement.

Kinetic Energy

Kinetic energy (KE) is the energy an object has due to its motion. If an object is moving, it has kinetic energy. The faster it moves, the more kinetic energy it possesses. The kinetic energy also depends on the object's mass. Here's how KE is calculated:\[KE = \frac{1}{2}mv^2\]This formula tells us that:

• The mass of the object (\( m \)) affects the kinetic energy linearly.
• The speed or velocity (\( v \)) of the object affects kinetic energy quadratically, meaning it has a squared effect.

In our scenario with the airplane having a velocity of 96.5 m/s, the kinetic energy was calculated as 8,679,871.125 Joules. This energy reflects how much work the airplane can do because of its motion.

Energy Conservation

Energy conservation is a fundamental principle in physics that helps us understand how energy remains constant in a closed system. It implies that energy cannot be created or destroyed, only transformed from one form to another. In terms of mechanical energy, this means the sum of an object's kinetic and potential energy remains the same, provided there is no external work done or energy lost. For the airplane scenario:

• The initial energy when the airplane was at rest is its potential energy at ground level—effectively zero, as it hasn't yet gained height or movement.
• After ascent and acceleration, its total energy is the sum of its gravitational potential energy and kinetic energy.

The changes calculated show a total mechanical energy change of approximately 52.9 million Joules. All this energy now accounts for its altitude and speed. Understanding energy conservation means recognizing this balance of energy forms, ensuring calculations align with the laws of physics.