Rita C.

asked • 06/15/14

An airplane whose speed in still air is 530 mph carries enough fuel for 10 hours of flight. On a certain flight it flies against a wind of 30mph. on the return

flight it travels with a wind of 30 mph. How far can the plane fly without refueling?

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Darlene N. answered • 06/15/14

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Hi, Rita!
 
The speed of the plane changes with the direction of the wind. So the speed on the original flight will actually be 500 mph, while the speed on the return will be 560 mph.
 
x = # hours on first flight
10-x = # hours on second flight
 
First flight miles = 500 • x
Second flight miles: 560(10-x)
 
The miles have to be the same on both flights, since the second is a return of the first.
 
500x = 560(10-x)
500x = 5600-560x
1060x = 5600
x = 5.283 hours
 
So the first flight, at the rate of 500 mph, lasted 5.283 hours. Multiply these and you'll get the distance of that first flight as 2641.509 miles.
 
Just checking on that second flight...it would last 10-5.283 hours, or 4.717 hours. Multiply that by the plane's speed of 560 mph, we get 2641.52 miles. Close enough to call it. :)
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Ashley W.

It might not be obvious why your first statement is true.  This has to do with inertial frames of reference, which I can't imagine anyone in an Algebra class has studied.  Basically, your speed relative to the ground (that is, your speed as seen by someone standing on the ground) will be different than your speed relative to something else, if that something else is moving relative to the ground.  For most purposes, you can just add your speed in one frame of reference, A, to the speed of that frame (A) in another frame of reference, B.
 
Planes fly in the air, literally; that is, the air isn't just the location of the plane, it is the entire frame of reference.  The point here is that if the air is moving, then the plane inside of it is moving along with it.  It is as if the air was a vehicle carrying the plane.  Therefore, if a plane's airspeed is fixed and the air isn't moving, then the plane's ground speed is also fixed. But, if the air starts moving relative to the ground, then the plane's ground speed (its speed relative to the ground) will change even though its airspeed (its speed relative to the air) hasn't changed.
 
Maybe an easier way to think about it is this: if you are walking 3 mph on a train going, say, 100 mph, then to someone seated on the train, you will be going +3 mph when you are walking toward the front of the train and -3 mph when you are walking toward the back of the train (since that person is also going 100 mph).  But, to someone on the ground, you would be going 103 mph when you are walking toward the front of the train (100 + 3) and 97 mph when you are walking toward the back of the train (100 - 3).  Your frame of reference is the train, which is moving, so your relative speed in another frame of reference (the ground's) is your speed relative to the train plus the train's speed relative to the ground.
 
Confused yet?  This is why they don't teach inertial frames of reference until college!
 
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06/17/14

Darlene N.

Ashley,
 
Your explanation of the first sentence is correct. However, as a teacher of everything from Algebra through AP Calc, I can tell you that these problems are discussed in Algebra and are worked as I described. Considering the "how far" question, this problem deals with distance relative to the ground. I saw no need to explain the inertial frames of reference since this problem can be worked without it.
 
Rita, if Ashley's explanation is helpful, then I'm glad she provided it. If it's confusing, don't worry -- the solution and explanation I provided are plenty for this problem.
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06/17/14

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