Chapter 6: Problem 56
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
Step by step solution
Understand the Energy Problem
Calculate the Work Done against Gravity
Find the Power Required for Climbing
Calculate the Fraction of Engine Power Used for Climbing
Result Interpretation
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with Vaia!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Power and Energy
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
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
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
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.