simple math question, d=rt

[Mavmb]

I saw this question on an airline pilot interview essay. If one plane is going .78 Mach and another plane is going .85 Mach and the planes are 50 miles apart, when will the faster plane catch up?

35 minutes, is that the right answer?

[fish314]

Well, there is a lot of information to answer this question that you don't have, but with some assumptions you can get an answer.

Assumptions:

The faster airplane is behind the slower airplane both on the same course. (Obviously, if they aren't going the same way, they might not ever meet!)

Mach*10= miles/minute. This is a pretty good approximation, but not exact. In order to get the exact you would need to know temperature, density, etc, etc, but since they didn't tell you the altitude or that the problem was on a standard day, this approximation is the best you've got.


Ok, on to the problem. The faster airplane is behind the slower airplane. The faster airplane is going .85 Mach and the slower airplane is going .78 Mach, so the faster airplane is gaining by .07 mach. That means the faster airplane is gaining by .7 miles/minute.

So how long does it take to close 50 miles at .7 miles/minute? 50/.7= 71.428 minutes, or about an hour and 11 minutes.

[Dugie8]

Originally Posted by fish314

Well, there is a lot of information to answer this question that you don't have, but with some assumptions you can get an answer.

Assumptions:

The faster airplane is behind the slower airplane both on the same course. (Obviously, if they aren't going the same way, they might not ever meet!)

Mach*10= miles/minute. This is a pretty good approximation, but not exact. In order to get the exact you would need to know temperature, density, etc, etc, but since they didn't tell you the altitude or that the problem was on a standard day, this approximation is the best you've got.


Ok, on to the problem. The faster airplane is behind the slower airplane. The faster airplane is going .85 Mach and the slower airplane is going .78 Mach, so the faster airplane is gaining by .07 mach. That means the faster airplane is gaining by .7 miles/minute.

So how long does it take to close 50 miles at .7 miles/minute? 50/.7= 71.428 minutes, or about an hour and 11 minutes.

ding ding ding, YOU WIN A CHEEEZBURGER!

[fish314]

I love cheeezburgers. Where do I collect? (Actually that's probably a really funny quote from a movie or TV or something, but I didn't recognize the reference).

[fish314]

Oh, hey mavmb, I think I figured out how you came up with 35 minutes, and it's actually a really common problem when people try to work out word problems.

Basically you lost track of the units, so instead of dividing 50 by .7, you probably tried to divide .7 by 50, and then maneuver the decimal place around so that the number made sense.

So here is how you avoid doing that: (BTW this works great in chemistry classes and physics classes if anyone is taking those).

Before you start working through a math problem, figure out what the units are in each of your different pieces, and what you expect the units of the answer to be.

For example, we had Mach numbers, which are unitless, and we had distance (measured in miles) and we knew that our answer had to be a time (minutes).

So far, nothing cancels, but if we had a conversion from Mach number to miles/minute, we might be able to get the "miles" in "miles/minute" to cancel with the "miles" in the distance, and leave us with just minutes.

Now, when you write out the problem, write it out with the units included, and make sure that the units make sense and come out correctly, and that will ensure that the answer comes out correctly also.

So if you had converted .07 Mach to .7 miles/minute and then divided by 50 miles, you would have noticed that the units of the answer was "1/minutes". Basically, the units are upside down. So that means the equation was probably upside down (or possibly screwed up some other way).

Now, this won't catch every mistake. But you'll catch a lot.

[tgrayson]

Originally Posted by fish314

Basically you lost track of the units, so instead of dividing 50 by .7, you probably tried to divide .7 by 50, and then maneuver the decimal place around so that the number made sense.

So complicated. Do you realize that .7 * 50 = 35 ?

[fish314]

Originally Posted by tgrayson

So complicated. Do you realize that .7 * 50 = 35 ?

Uhh. yeah. That's what he did.....

Whatever, the units don't work out that way either-- That way you get miles squared/minute, not minutes.

[tgrayson]

Originally Posted by fish314

Whatever, the units don't work out that way either-- That way you get miles squared/minute, not minutes.

I have yet to see non-technical types that understand this line of argumentation. The idea that units can cancel with other units is mind-boggling.

[Mavmb]

Okay thanks! I see what I did. Doh!

[NJA_Capt]

Originally Posted by fish314

So how long does it take to close 50 miles at .7 miles/minute? 50/.7= 71.428 minutes, or about an hour and 11 minutes.

At .91 you could make up the distance in :38 mins.

[fish314]

Originally Posted by NJA_Capt

At .91 you could make up the distance in :38 mins.

Dude, you forgot the big arrow at the bottom to point to your avatar!

Like this:


<----------------------------------------

[aerospacepilot]

1 Hour, 3 Minutes, 11 Seconds assuming the aircraft are at 30,000ft.

Find the speed of sound at altitude
Calculate the velocities of the two jets
Calculate the speed the faster one overtakes the smaller one
Time = Distance / Velocity

Takes less than a minute to figure out. Not quite as quick as the "shortcut" mentioned above, but this way is 100% accurate. The other way was off by 12%, and that could make a difference if we are talking hours.

[fish314]

Originally Posted by aerospacepilot

1 Hour, 3 Minutes, 11 Seconds assuming the aircraft are at 30,000ft.

Find the speed of sound at altitude
Calculate the velocities of the two jets
Calculate the speed the faster one overtakes the smaller one
Time = Distance / Velocity

Takes less than a minute to figure out. Not quite as quick as the "shortcut" mentioned above, but this way is 100% accurate. The other way was off by 12%, and that could make a difference if we are talking hours.

I think you may have made a math error, but you're correct that your method would give the exact number.

(Although the original problem didn't include an altitude, so how did you know to use 30,000 ft? If you compute at sea level, or 20,000 or 50,000 you'll get different times)

Using the same 30,000 feet, though, I come up with:

a (the speed of sound on a standard day)=994.8 ft/sec

a*3600 seconds/hour=3581280 ft/hour

3581280 ft/hour / (6076 ft/NM)= 589.41 NM/hour

The slow airplane's speed =.78 Mach =.78*589.41= 459.7 NM/hour

The fast airplane's speed= .85 Mach = .85*589.41= 501.0

The overtake is the fast speed minus the slow speed =501-459.7= 41.3NM/Hour

50 Miles/(41.3 NM/hour)=1.211 hours

This is the same as 1 hour and an additional .211 hours*60 minutes/hour or 1 hour and 12.66 minutes.

Or 1 hour 12 minutes, 39.6 seconds.

The answer that I came up with using my WAG was 71.428 minutes, which is the same as 1.190 hours. Therefore the error in that method isn't 12%, it's actually only (1.211-1.190)/1.211=.0173, or 1.73%

So in short, it was a very good approximation

[fish314]

I also noticed that the original problem didn't specify whether the two airplanes were 50 Nautical miles apart or 50 statute miles apart, but I assumed they were 50 Nautical miles apart.

The difference? Well a statute mile is 5280 ft. A Nautical mile is 6076 feet. So if the airplanes were 50 statute miles apart, they were 264000 feet apart. If they were 50 nautical miles apart, they were 303800 feet apart.

That's a difference of 39800 ft, which is 6.55 NM, or 7.54 statute miles. At about 7 miles/minute, that's about 1 minutes worth of error (close to 1%) right there.