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Chapter 26: Problem 2

A small airplane has an airspeed (speed with respect to air) of \(200 \mathrm{~km} / \mathrm{h}\). Find the time for the airplane to travel \(1000 \mathrm{~km}\) if there is (a) no wind, (b) a headwind of \(35 \mathrm{~km} / \mathrm{h},\) and \((\mathrm{c})\) a tailwind of \(35 \mathrm{~km} / \mathrm{h}\).

Short Answer

Expert verified
(a) 5 hours, (b) approximately 6.06 hours, (c) approximately 4.26 hours.

Step by step solution

01

Identify the airspeed of the airplane

The problem states that the airplane has an airspeed of \( 200 \, \mathrm{km/h} \). This is the speed of the airplane relative to the air.
02

Calculate travel time with no wind

In the absence of wind, the ground speed of the airplane is equal to its airspeed, \( 200 \, \mathrm{km/h} \). To find the time \( t \) it takes to travel \( 1000 \, \mathrm{km} \), use the formula \( t = \frac{\text{distance}}{\text{speed}} \). Therefore, \( t = \frac{1000}{200} = 5 \, \mathrm{hours} \).
03

Calculate travel time with a headwind

When there is a headwind, it slows down the airplane relative to the ground. The ground speed will be the airspeed minus the headwind speed: \( 200 \, \mathrm{km/h} - 35 \, \mathrm{km/h} = 165 \, \mathrm{km/h} \). Again, use the formula \( t = \frac{\text{distance}}{\text{speed}} \): \( t = \frac{1000}{165} \approx 6.06 \, \mathrm{hours} \).
04

Calculate travel time with a tailwind

With a tailwind, the airplane is assisted in moving forward. The ground speed will be the airspeed plus the tailwind speed: \( 200 \, \mathrm{km/h} + 35 \, \mathrm{km/h} = 235 \, \mathrm{km/h} \). Use the formula \( t = \frac{\text{distance}}{\text{speed}} \): \( t = \frac{1000}{235} \approx 4.26 \, \mathrm{hours} \).

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Key Concepts

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

Airspeed Calculation
Understanding how to calculate the airspeed of an airplane is crucial for tackling many physics and navigation problems. Airspeed refers to how fast an airplane travels relative to the surrounding air. It is an important measure because it directly affects the lift force acting on the airplane, which is needed to keep it flying.
  • Airspeed vs. Ground Speed: While airspeed measures speed relative to air, ground speed relates to how fast an airplane is moving over the ground surface. These two measures can differ significantly due to the presence of wind.
  • Formula: Airspeed is typically measured in kilometers per hour (km/h) or knots. In this exercise, we are dealing with an airspeed of \(200\, \mathrm{km/h}\), which we use as a reference point for further calculations.
Knowing the airspeed helps you determine how an airplane interacts with wind conditions, which ties into our next concept.
Wind Effect on Speed
Wind has a significant effect on the speed of an airplane. It can either slow down the plane (headwind) or speed it up (tailwind). Understanding wind effects is vital for predicting accurate travel times and navigation.
  • Headwind: A headwind opposes the aircraft's direction of travel. This reduces the ground speed. For example, with a headwind of \(35\, \mathrm{km/h}\), the plane's effective speed becomes \(200 \mathrm{km/h} - 35 \mathrm{km/h} = 165 \mathrm{km/h}\).
  • Tailwind: Conversely, a tailwind assists the plane by moving in the same direction. This increases the ground speed. So if we have a tailwind of \(35\, \mathrm{km/h}\), the plane's ground speed rises to \(200 \mathrm{km/h} + 35 \mathrm{km/h} = 235 \mathrm{km/h}\).
By adjusting the airspeed based on the wind, navigators can provide more accurate estimates for travel duration.
Travel Time Calculation
Once you have the speed of the airplane relative to the ground, calculating travel time is essential to understanding how long the journey will take under different conditions. The formula used here is:
\[t = \frac{\text{distance}}{\text{speed}}\]This formula helps us determine how long it will take to travel a specific distance (in this case, \(1000 \mathrm{km}\)).
  • No Wind: The airplane's ground speed is the same as its airspeed. Calculation: \(t = \frac{1000}{200} = 5 \, \mathrm{hours}\).
  • With Headwind: The ground speed decreases due to opposing wind, leading to more travel time: \(t = \frac{1000}{165} \approx 6.06 \, \mathrm{hours}\).
  • With Tailwind: The airplane benefits from the wind, reducing travel time: \(t = \frac{1000}{235} \approx 4.26 \, \mathrm{hours}\).
Accurate travel time calculations depend on understanding both speed and the impact of environmental factors like wind.

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