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Chapter 7: Problem 61

When a small plane flies with the wind, it can travel 800 miles in 5 hours. When the plane flies in the opposite direction, against the wind, it takes 8 hours to fly the same distance. Find the average velocity of the plane in still air and the average velocity of the wind.

Short Answer

Expert verified
The average velocity of the plane in still air is 130 mph and the average velocity of the wind is 30 mph.

Step by step solution

01

Define the Variables

Let \(p\) be the speed of the plane in still air and \(w\) be the speed of the wind. When the plane flies with the wind, its effective speed is \(p + w\) and when it flies against the wind its effective speed is \(p - w\).
02

Formulate Equations

Using the formula for speed, which is distance divided by time, we can create two equations from the given conditions: \[p + w = 800 / 5\] and \[p - w = 800 / 8\].
03

Solve the Equations

First, solve these equations to find \(p\) and \(w\). By adding these two equations:\[(p + w) + (p - w) = (800 / 5) + (800 / 8)\]. Simplify to get:\[2p = 160 + 100\], which implies \(p = 130\). Substitute \(p = 130\) into the equation \(p + w = 800 / 5\) to get \(w = 30\).

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Key Concepts

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

Average Velocity
Average velocity helps us understand how fast an object is moving over a specific time, regardless of any stops or changes in speed along the way. In the context of our problem, we are trying to find the plane's average velocity in still air and the wind鈥檚 average velocity over the entire journey.
To calculate average velocity, use the formula:
  • Average Velocity = Total Distance / Total Time.
From the problem, flying 800 miles with the wind takes 5 hours, and flying back against the wind takes 8 hours, covering another 800 miles. Therefore, when calculating each portion separately, you can determine the average velocity over multiple legs of the journey to find distinct velocities for components like wind or still air.
Speed Calculation
Speed is the measurement of how quickly an object travels from one point to another. In the exercise, calculating the speed involves determining how fast the plane flies both with and against the wind.
By following step-by-step calculations:
  • Calculate the speed with the wind: Since the plane travels 800 miles in 5 hours, the speed with the wind is 800 divided by 5, giving us 160 mph.
  • Speed against the wind shows us how the wind velocity affects travel time. Here, the plane covers the same distance in 8 hours, so the speed is 800 divided by 8, equating to 100 mph.
These calculations lay the foundation for solving problems involving different speeds due to varying conditions.
Algebraic Equations
Algebraic equations play a crucial role in solving the problem by representing real-world situations mathematically.
Define variables for an easy representation:
  • Let \(p\) represent plane speed in still air.
  • Let \(w\) represent wind speed.
Using algebra, we set up these equations based on speed calculations:
  • The equation \(p + w = 800 / 5\) represents total speed with the wind.
  • The equation \(p - w = 800 / 8\) represents total speed against the wind.
Combining these equations systematically like shown in the solution helps in isolating variables, making it possible to find the separate values for \(p\) and \(w\). This method is highly useful in various practical applications where direct measurements are not feasible.
Wind Velocity
Wind velocity refers to the speed and direction of wind relative to a moving object. In the exercise, it affects the plane's travel time and effective speed.
Consider its effect in both directions:
  • When traveling with the wind, its velocity adds to the plane's speed, allowing it to cover more distance in less time.
  • When traveling against the wind, its velocity reduces the plane's speed, increasing travel time for the same distance.
After solving the equations, the wind velocity \(w\) was found to be 30 mph. Understanding wind velocity is essential, especially for navigation and flight planning where external conditions significantly impact travel efficiency.

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