Question

An 1865-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 is \(53,021,175.825 \text{ J}.\)

Step-by-step solution

Identify Given Information

We have the mass of the airplane, \( m = 1865 \) kg, the initial altitude \( h_1 = 0 \) m, the final altitude \( h_2 = 2420 \) m, and the velocity \( v = 96.5 \) m/s at cruising altitude.

Calculate Initial Mechanical Energy

The initial mechanical energy consists of initial gravitational potential energy and initial kinetic energy. Since the airplane starts at rest and at ground level, both of these energies are zero:\( E_{initial} = mgh_1 + \frac{1}{2}mv_1^2 = 1865 \times 9.81 \times 0 + \frac{1}{2} \times 1865 \times 0^2 = 0. \)

Calculate Final Mechanical Energy

The final mechanical energy consists of gravitational potential energy at cruising altitude and kinetic energy due to the constant speed:\[ E_{final} = mgh_2 + \frac{1}{2}mv^2 = 1865 \times 9.81 \times 2420 + \frac{1}{2} \times 1865 \times (96.5)^2. \] Calculating these gives us the final mechanical energy.

Calculate Change in Mechanical Energy

The change in mechanical energy is the difference between final and initial mechanical energy:\[ \Delta E = E_{final} - E_{initial}. \]Since \( E_{initial} = 0 \), \( \Delta E = E_{final}. \)

Substitute Values and Solve

Substitute the known values into the equations and solve:\[ E_{final} = 1865 \times 9.81 \times 2420 + \frac{1}{2} \times 1865 \times 96.5^2 = 44,344,413 + 8,676,762.825 = 53,021,175.825 \text{ J}. \]So, \( \Delta E = 53,021,175.825 \text{ J}. \)

Key concepts

Gravitational Potential Energy

Gravitational potential energy is a type of energy that an object possesses due to its position relative to the Earth. In simpler terms, it's the energy held by an object because of its height above the ground. The higher the object, the more gravitational potential energy it holds. This can be calculated using the formula:\[ GPE = mgh \]

• Where \( m \) is the mass of the object in kilograms,
• \( g \) is the acceleration due to gravity (approximately \( 9.81 \text{ m/s}^2 \) on Earth),
• \( h \) is the height above the reference point, typically the ground, in meters.

As an example from the exercise, when the airplane climbs to an altitude of 2420 meters, its gravitational potential energy increases. This is because it gains height.The formula allows us to compute this energy increase by substituting in the elevation (2420 m), the mass of the plane (1865 kg), and the gravitational constant \( g \). The resulting energy is the gravitational contribution to the airplane's mechanical energy.

Kinetic Energy

Kinetic energy, on the other hand, is all about the energy of motion. It's the energy that an object has because it's moving. The faster the object moves, the more kinetic energy it has. This is calculated with:\[ KE = \frac{1}{2}mv^2 \]

• Where \( m \) again is the mass of the object in kilograms,
• \( v \) is the velocity or speed of the object in meters per second.

For the airplane in the example, once it reaches a cruising speed of 96.5 m/s, it's moving fast enough to possess a significant amount of kinetic energy. By inserting this speed and the airplane's mass into the formula, we calculate the kinetic energy portion of the airplane's mechanical energy. Together with gravitational potential energy, it tells us how much total energy the airplane holds while flying.

Energy Conservation

The principle of energy conservation states that energy cannot be created or destroyed, only transformed from one form to another. This is crucial in solving problems involving mechanical energy, like the airplane example.
In our scenario, the mechanical energy change in the airplane is the sum of its kinetic and gravitational potential energy changes. Because it starts at rest, its initial energy is zero. As it climbs, the airplane's potential energy increases due to elevation and its kinetic energy due to its speed.
Energy conservation shows us that the work done on the airplane to gain this new height and speed is equal to the mechanical energy it has accumulated. This means the calculated change in energy reflects the work done against gravity and to accelerate it to its final speed. Thus, in the absence of losses, the total energy remains constant before and after the airplane climbs.