The average airspeed of a singleengine aircraft is 150 miles...

Name: Problem 64, Chapter 10: Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry
Uploaded: 2021-06-28T13:31:58-08:00
Duration: 4 min 52 s
Channel: Andrija Isakov
Description: The average airspeed of a singleengine aircraft is 150 miles per hour. If the aircraft flew the same distance in 2 hours with the wind as it flew in 3 hours against the wind, what was the wind speed?

The average airspeed of a singleengine aircraft is 150 miles per hour. If the aircraft flew the same distance in 2 hours with the wind as it flew in 3 hours against the wind, what was the wind speed?

Key Concepts

Distance-Speed-Time Relationship

This fundamental relationship, expressed as distance equals speed multiplied by time, is essential in solving motion problems. It allows one to set up equations that relate the known variables to the unknown, such as determining wind speed when distances for both cases are equal.

Setting Up and Solving Equations

In problems where an unknown value, like wind speed, affects the outcome, setting up equations based on the conditions given (such as equal distances traveled in different times) is key. Solving these equations involves algebraic manipulation to isolate the unknown variable.

Relative Velocity

Relative velocity describes how the speed of an object is affected by the movement of the medium through which it travels. In aviation, this concept is crucial because an aircraft's speed relative to the ground (ground speed) is influenced by wind, adding to or subtracting from its airspeed.

Wind Effects on Ground Speed

The impact of wind on an aircraft's speed can be observed in scenarios involving tailwinds and headwinds. A tailwind increases the aircraft's ground speed by adding to its airspeed, while a headwind decreases the ground speed by subtracting from the airspeed.

Transcript

00:01 In this question, we're talking about systems of linear equations.

00:06 We have a situation where an airplane whose airspeed is 150 miles per hour, so the speed due to just the engine and just the airplane is 150, and it travels a certain distance in three hours if it goes against the wind, and it travels that same distance, we can think about going on a round trip, going back the same distance in two hours with the wind.

00:43 We want to find how fast this wind is.

00:47 So when the airplane is going against the wind, then the overall speed that the airplane will be going at, the ground speed, will be less than its original airplane speed, and with the wind, it's obviously more.

01:04 And how much less and how much more is given by the wind speed? if the wind is blowing a certain number of miles per hour, then that's how much faster the airplane will go if it's with the wind.

01:15 Okay, so let's give a name to this variable that we want to find.

01:21 So we want to find x, the speed of the wind.

01:27 And we also have another unknown variable, the distance.

01:31 Although, as we'll see later, we might not need to know its value or give it a variable name in order to solve for x, which is what we'll see.

01:40 You're interested in.

01:43 So like we said, the plane will go the wind speed faster than its original speed if it's going with the wind.

01:52 So the speed with the wind is 150 plus x.

01:56 No speed is distance divided by time.

01:59 So we can write that that equals the distance it travels, which we can give a variable name after all, d divided by the time, which is two hours, two hours with the wind.

02:11 And now we can form another equation with the against wind information.

02:21 So since we have two unknowns, we need two equations to solve for either of them.

02:27 So this equation is similar.

02:30 Since we're going against the wind, the x is subtracted.

02:34 So because we're going that much slower and recovering that distance in three hours...

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