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When its 75-kW (100-hp) engine is generating full power, a small single-engine airplane with mass 700 kg gains altitude at a rate of 2.5 m/s (150 m/min, or 500 ft/min). What fraction of the engine power is being used to make the airplane climb? (The remainder is used to overcome the effects of air resistance and of inefficiencies in the propeller and engine.)

Short Answer

Expert verified
22.89% of the engine power is used for climbing.

Step by step solution

01

Understand the Energy Problem

The airplane's engine produces a maximum power output of 75 kW. We'll need to find out the share of this power that is used to generate the airplane's altitude climb.
02

Calculate the Work Done against Gravity

The work done to lift something to a certain height is determined by the weight of the object and the height it is lifted. Here, the airplane, with a mass of 700 kg, is climbing vertically at 2.5 m/s. First, calculate the gravitational force \( F = m \cdot g = 700 \text{ kg} \times 9.81 \text{ m/s}^2 \), which gives \( F = 6867 \text{ N} \).
03

Find the Power Required for Climbing

Power is the rate at which work is done. Here, power required for climbing is \( P = F \times v = 6867 \text{ N} \times 2.5 \text{ m/s} \). This results in \( P = 17167.5 \text{ W} = 17.1675 \text{ kW} \).
04

Calculate the Fraction of Engine Power Used for Climbing

The fraction of the engine power used for climbing is calculated by comparing the power used for climbing with the total power available. This is \( \frac{17.1675 \text{ kW}}{75 \text{ kW}} \). Perform the division to find the fraction.
05

Result Interpretation

Calculating, \( \frac{17.1675 \text{ kW}}{75 \text{ kW}} \approx 0.2289 \), which means approximately 22.89% of the engine's power is used for climbing.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power and Energy
Power and energy are fundamental concepts in physics that help us understand how machines and vehicles like airplanes function. Power is defined as the rate at which work is done or energy is transferred over time. It is measured in watts (W). In the context of our problem, the engine of the airplane generates a power output of 75 kW to make the airplane climb.
The total energy required for the airplane to climb can be calculated using the forces acting on it as it gains altitude. By knowing the power generated and the efficiency, we can determine how much of that power is used to overcome other forces like gravity and air resistance. This way, we learn not only about the energy input but also the different components consuming it.
Understanding power and energy in mechanics provides insight into the efficiency and performance of engines, allowing us to calculate and optimize how much energy is used by different parts of the system.
Mechanics
Mechanics is the branch of physics concerning the motion of objects and the forces acting on them. When examining the mechanics of an airplane, we assess various factors like mass, velocity, and forces. In our problem, mechanics helps us calculate the work done by the engine to lift the airplane against the gravitational force.
The fundamental equation used here is the work-energy principle: work done equals force times distance or, in terms of power, force times velocity. The gravitational force, which is the weight of the airplane, acts against the airplane's movement upward. By multiplying this force with the airplane's vertical velocity, we find the required power to achieve this climb rate. Understanding the mechanics involved is crucial for determining the efficiency of power use in overcoming resistance forces in climbing and moving an aircraft.
Airplane Dynamics
Airplane dynamics involves understanding how forces affect the motion of an aircraft. In this problem, an airplane's climbing rate is influenced by various forces, including lift produced by the wings, thrust from the engine, drag from air resistance, and gravity acting downwards.
The dynamics necessitate examining how power produced by the engine is divided between different tasks such as providing lift to counteract gravity and overcoming drag to maintain speed. In our question, we focus on the part of the power that specifically contributes to climbing. The rest of the power compensates for drag and inefficiencies. By breaking down these dynamics, students can better grasp how planes achieve sustained flight and safe maneuvers, showing them real-world applications of physics principles.
Engine Efficiency
Engine efficiency determines how effectively an engine converts fuel into work, impacting how much energy is available for motion. It is a measure of how much power is lost in unwanted forms like heat and friction versus how much is actually used for its intended function.
In our problem, we find that about 22.89% of the engine's power output is used specifically for climbing, implying that the remaining power deals with overcoming non-beneficial forces such as air resistance or inefficiencies. Improving engine efficiency would mean utilizing a higher percentage of the engine's power for useful work, such as climbing or cruising.
Understanding engine efficiency is important in designing and operating airplanes, as it influences fuel consumption, costs, and environmental impact. It’s a crucial consideration for aerospace engineering, striving for improvements in performance and sustainability.

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Most popular questions from this chapter

All birds, independent of their size, must maintain a power output of 10\(-\)25 watts per kilogram of body mass in order to fly by flapping their wings. (a) The Andean giant hummingbird (\(Patagona gigas\)) has mass 70 g and flaps its wings 10 times per second while hovering. Estimate the amount of work done by such a hummingbird in each wingbeat. (b) A 70-kg athlete can maintain a power output of 1.4 kW for no more than a few seconds; the \(steady\) power output of a typical athlete is only 500 W or so. Is it possible for a human-powered aircraft to fly for extended periods by flapping its wings? Explain.

A little red wagon with mass 7.00 kg moves in a straight line on a frictionless horizontal surface. It has an initial speed of 4.00 m/s and then is pushed 3.0 m in the direction of the initial velocity by a force with a magnitude of 10.0 N. (a) Use the work\(-\)energy theorem to calculate the wagon's final speed. (b) Calculate the acceleration produced by the force. Use this acceleration in the kinematic relationships of Chapter 2 to calculate the wagon's final speed. Compare this result to that calculated in part (a).

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An object has several forces acting on it. One of these forces is \(\overrightarrow{F}= axy\hat{\imath}\), a force in the \(x\)-direction whose magnitude depends on the position of the object, with \(\alpha = 2.50 \, \mathrm{N/m}^2\). Calculate the work done on the object by this force for the following displacements of the object: (a) The object starts at the point (\(x = 0\), \(y = 3.00\) m) and moves parallel to the x-axis to the point (\(x= 2.00\) m, \(y = 3.00\) m). (b) The object starts at the point (\(x = 2.00\) m, \(y = 0\)) and moves in the \(y\)-direction to the point (\(x = 2.00\) m, \(y = 3.00\) m). (c) The object starts at the origin and moves on the line \(y = 1.5x\) to the point (\(x = 2.00\) m, \(y = 3.00\) m).

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