Question

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

22.89% of the engine power is used for climbing.

Step-by-step solution

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.

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 $F=6867\text{ N}$.

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 $P=17167.5\text{ W}=17.1675\text{ kW}$.

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.

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.

Key concepts

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.