Question 543345: A plane travels 450 mph for 1000 miles with the wind in 30 minutes less time than against the wind. what is the speed of the wind?
Answer by lmeeks54(111)   (Show Source):
You can put this solution on YOUR website! This problem revolves around the simple relationship between distance, time, and speed (called rate). But it is a little complicated by the effects of wind, which is what we are asked to solve for.
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The basic formula is: d = t * r, where d = distance, t = time, and r = rate. Note, another way to get tripped up in problems such as this is in comparing units of distance, time, or speed. For example, d is often in miles or kilometers, and speed is often in mph or kph; these go together nicely, and in this instance, you would think time would be in hours. However, we are sometimes given other information to help solve the problem, such as the difference in times between the downwind and upwind flights is 30 minutes. Not a real big deal, but it is important to watch the units so they all match.
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Let's consider this as two trips and trip to keep them separate (for clarity), but find the ways that relate them (for the solution).
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Let d = distance (this is given as 1,000 miles)
Let t = time of either trip
Let r = the speed of the aircraft in still air (we'll assume that is constant)
Let w = the speed of the wind, either pushing or slowing the aircraft (and we'll assume that is constant too)
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Remember, d = r * t or, written another way, t = d / r
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Trip 1, downwind leg:
given:
d = 1,000
we don't know r or w, but we know combined speed (the speed the aircraft makes helped by the wind):
r + w = 450
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we can calculate t for the downwind trip:
t = d / (r + w)
t = 1,000 miles / 450 mph
t = 2.22 hrs
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Trip 2, upwind leg:
the upwind trip is the same d = 1,000 miles and it takes 30 minutes longer. So, let's convert t = 2.22 hrs into t = 133.33 mins by multiplying by 60 mins/hr for the downwind trip so we can add the delay caused by the headwind.
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t upwind = 133.33 + 30 = 163.33 mins
Let's convert the t upwind back into hrs by dividing by 60 mins/hr:
t = 2.72 hrs
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Now we can go back into our basic formula, this time solving for speed of the aircraft on the upwind leg, considering both, r, the constant speed of the aircraft, and the slowing affects of the wind, w.
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speed = d / t, recall in this case, speed = r - w
speed = 1,000 miles / 2.72 hrs
speed = 367.35 mph
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Now we have two speeds:
Downwind, r + w = 450 (given)
Upwind, r - w = 367.35 (calculated)
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We now need to break out r (aircraft speed in still air) and w (constant wind speed, that either helps or hurts aircraft speed over the ground):
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Consider this: calculate the difference in speeds downwind and upwind:
450 mph - 367.35 mph = 82.65 mph
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This equals the effects of the wind on the aircraft. Now some of this 82.65 mph is used up on the downwind leg and some on the upwind leg. If we can assume the wind speed is constant (we have to, or else we'd need other information) then we can divide the 82.65 mph of wind by 2 (for two legs).
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Wind effect downwind is +82.65/2 and wind effect upwind is -82.65/2
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The answer to your problem is w = 41.33 mph
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Cheers,
Lee
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PS if you wanted to know the speed of the aircraft in still air, subtract the wind speed of 41.33 mph from the given combined speed (downwind) of 450 mph: 450 ph - 41.33 mph = 408.67 mph or, add it to the calculated upwind speed of 367.35: 367.35 mph + 41.33 mph = 408.67 mph.
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