City A lies directly west of city B . When there is no wind,...

City A lies directly west of city $B$ . When there is no wind, an airliner makes the 5550 -kmround-trip fight between them in 6.60 $\mathrm{h}$ of flying time while traveling at the same speed in both directions. When a strong, steady $225-\mathrm{km} / \mathrm{h}$ wind is blowing from west to east and the airliner has the same airspeed as before, how long will the trip take?

City A lies directly west of city $B$ . When there is no wind, an airliner makes the 5550 -kmround-trip fight between them in 6.60 $\mathrm{h}$ of flying time while traveling at the same speed in both directions. When a strong, steady $225-\mathrm{km} / \mathrm{h}$ wind is blowing from west to east
and the airliner has the same airspeed as before, how long will the trip take?

Key Concepts

Airspeed versus Groundspeed

This concept differentiates between the speed of an aircraft relative to the surrounding air (airspeed) and its speed over the ground (groundspeed). When wind is present, the aircraft’s groundspeed is the vector sum of its airspeed and the wind’s speed, which means that in the direction of the wind the groundspeed increases and in the opposite direction it decreases.

Wind Influence on Flight Time

Wind can significantly alter the effective speed of an aircraft along its path. A tailwind increases groundspeed and decreases travel time, while a headwind does the opposite. Understanding the impact of wind on flight duration requires calculating the effective speed during each leg of a journey and then applying the time, distance, and speed relationship.

Time-Distance-Speed Relationship

This fundamental concept relates time, distance, and speed using the equation time = distance/speed. In problems involving variable speeds, such as those affected by wind, it is important to calculate the time for each segment of the trip separately and then sum them to determine the total travel time.

Transcript

00:01 All right, in this problem, we have two cities.

00:03 We have city a and we have city b.

00:07 And we're given some information about the cities.

00:09 If you do a round trip, the total distance for that round trip, that's d sub t, is equal to 5 ,550 kilometers.

00:21 And for this particular airliner that we're dealing with, the time to do that total trip is 6 .6 hours.

00:29 And the question is, if you had a wind, wind coming out of the west at 225 kilometers per hour, which is a little bit unrealistic.

00:42 It's about 140 miles an hour.

00:44 But if you had that wind with that same airliner, how does that affect the total time for the round trip of that aircraft? okay, one thing to realize is that this is a relative motion problem because you have an object, in this case, say airplane that's moving, but the thing that it's moving on or through, in this case the air is also moving.

01:11 And whenever you have that, it's really a vector problem that the total velocity of the object in question, again, the airplane, is equal to the velocity of the object.

01:24 Plus the velocity of the medium.

01:28 And i'll just call those v -o and vm.

01:31 You can do this problem without treating it as vectors, but then when you go ahead and have some kind of a sidewind instead of just a head wind or tailwind, it becomes harder to solve.

01:43 So if we think about this as a vector problem, we think about it as a relative motion problem.

01:48 That's pretty easy.

01:50 Now, if we think about it, when we are moving from a to b, moving from a to b, we've got a tailwind.

01:59 And so numerically, that's going to just be the velocity of the airplane plus the velocity of the wind.

02:08 And we add those two things.

02:10 And that's for the velocity from a to b.

02:12 When we're going the other direction, the velocity from b to a, that again is going to be the velocity of the airplane in this question, but we're going to have to subtract the wind.

02:28 Another way to look at that is that this is going to be that vector plus the wind vector.

02:34 This is going to be the airplane vector and then plus the wind vector, but the wind vector is going the other way.

02:41 So we subtract them.

02:43 So really the big thing we need to do is to be able to solve for that speed of the airplane and then go from, uh, go from there.

02:52 Just a quick excite.

02:53 Many of you know that d is equal to rt.

02:57 I remember it as dirt.

02:59 The distance is equal to the rate, in this case the velocity times the time.

03:05 If you don't have that equation memorized, a very easy way to do it is to realize that the speed, which in this case is kilometers per hour, has to equal the distance divided by the time.

03:17 Kilometers per hour has to equal kilometers per hour...

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