# Average Speed

#### pavelump

##### Well-Known Member
If you go from point A to B at a GS of 200 kts. and then back from B to A at 100 kts what is your average speed?

I've heard that this is an interview question and that the answer is 133, but I haven't figured out how that number is arrived at.

Dave

#### flyitup

##### Well-Known Member
I would think the answer would be 150kts...

200kts there and 100kts back would average out to be 150kts...

Then again I'm not exactly the "engineer type". Perhaps one of our engineering JC members will stop by to put in their .02 (braidkid)! #### EatSleepFly

##### Well-Known Member
Eh, it has something to do with the fact that over a given distance you spend longer going slower. I can visualize it in my head, but I'm still trying to figure out how to show it mathematically.... #### Derg

##### Cap, Roci
Staff member
Well, let's say the distance from point a to point b is 100 miles.

a-b would take you .5 hours
b-a wouild take you 1.0 hours

So you traveled a total distance of 200 miles of which you flew 1.5 hours.

distance = rate x time

200 miles = rate x 1.5 hours

rate = 200 miles / 1.5 hours

rate = 133.333 miles/hour

#### EatSleepFly

##### Well-Known Member
OK, got it....now lets see if I can explain it:

For the sake of discussion, we'll say that the distance between points A and B is 400nm.

At a GS of 200 kts., it will take you 2 hrs. to cover those 400nm.

For the return trip at a GS of 100 kts., it will take you 4 hours.

From there, all you have to do is remember the equation for average speed, which is:

Avg. Speed = Distance Traveled/Time Traveled

so...

Avg. Speed = 800nm/6 hrs.

= 133 kts.

Hows that?

#### pavelump

##### Well-Known Member
OK, nevermind I just figured it out.

Here's what you do. What we're looking for is the harmonic mean (sounds like I know what I'm talking about doesn't it?).

Basically it's based on the rt=d formula. The only thing is, we're not given enough info so we have to come up with an arbitrary number for d (distance). So:

r1 = 200 kts
r2 = 100 kts
d = 500 nm
Let the times be t1 &amp; t2
t = total time (t1+t2)
Our problem is to find the rate (r) for travel of the distance 2d

t1 (A to B) = d/r1
t1 = 500nm/200kts
t1 = 2.5 hrs

t2 (B to A) = d/r2
t2 = 500nm/100kts
t2 = 5 hrs

So, to find the mean (average) of the two:

(r)(t) = 2d
(r)(t1+t2) = 2d
r (2.5 hrs + 5 hrs) = 2(500nm)
r * 7.5 hrs = 1000nm
r = 1000nm/7.5hrs
133.333333 kts = r

thank god for the internet. I can only take partial credit. I found the solution here: http://jwilson.coe.uga.edu/emt725/Average.Rate/Average.Rate.html

Phew, now I can go to bed.
Dave

#### EatSleepFly

##### Well-Known Member
Guess I type too slow #### pavelump

##### Well-Known Member
freaky, we were all coming to the same conclusion simultaneously....

yipes

#### Derg

##### Cap, Roci
Staff member
Ha! Had all of you kids beat by 4 minutes!

How's that for a "C" average Air Science student from 'Riddle!

curses...

#### flyitup

##### Well-Known Member
[ QUOTE ]
Guess I type too slow [/ QUOTE ]

Guess I think too slow... #### E_Dawg

##### Moderator
That's one of the crap things about flying... if there is ANY wind... it slows you down if you're making a round trip.

#### aloft

##### New Member
No fair, you're playing algebraic games in there...

#### Derg

##### Cap, Roci
Staff member
Yeah Aloft, but do I get any credit for the shortest approach? #### reaperman

##### Well-Known Member
Of course you do. But the most credit goes for doing it entirely with arithmetic - no math. Math has no place in a cockpit. The general mathematical solution is nearly as short but, as I said, should not be done while operating heavy machinery.

#### carlos

##### Well-Known Member
[ QUOTE ]
Of course you do. But the most credit goes for doing it entirely with arithmetic - no math. Math has no place in a cockpit. The general mathematical solution is nearly as short but, as I said, should not be done while operating heavy machinery.

[/ QUOTE ]

I don't get the math versus arithmetic point (arithmetic is a type of math). How Doug figured it is the simplest quickest means of doing it while operating heavy machinery or otherwise. Using a specific simple concrete example with round numbers (100 miles distance, for example) to solve a general problem is the best way to quickly compute. No reason to make it harder than it is.

#### pavelump

##### Well-Known Member
[ QUOTE ]
No reason to make it harder than it is.

[/ QUOTE ]

what are you talking about? I didn't even mention the ever popular

r = 2(r1)(r2)/r1+r2 Dave

#### reaperman

##### Well-Known Member
[ QUOTE ]
arithmetic is a type of math

[/ QUOTE ] Incorrect. Number theory is a type of mathematics. But we aren't talking about that here.

[ QUOTE ]
How Doug figured it is the simplest quickest means of doing it while operating heavy machinery or otherwise.

[/ QUOTE ] That's the point of what we've all been saying. If you're going to disagree with me, you at least have to contradict me.

[ QUOTE ]
Using a specific simple concrete example with round numbers (100 miles distance, for example) to solve a general problem is the best way to quickly compute.

[/ QUOTE ]You can't solve a general problem with specific numbers, only the specific problem.

Of course Doug really did do the math -- at some time in the dim and distant past, then internalized the lesson. The lesson being that time is the denominator and must be honored always.

See pavelump's reply for the math.

#### carlos

##### Well-Known Member
[ QUOTE ]
[ QUOTE ]
arithmetic is a type of math

[/ QUOTE ] Incorrect. Number theory is a type of mathematics. But we aren't talking about that here.

[/ QUOTE ]

Perhaps "type" was the wrong word. From "The American Heritage Dictionary of the English Language":
"arithmetic. The mathematics of integers under addition, subtraction, multiplication, division, involution, and evolution."

[ QUOTE ]
[ QUOTE ]
How Doug figured it is the simplest quickest means of doing it while operating heavy machinery or otherwise.

[/ QUOTE ] That's the point of what we've all been saying. If you're going to disagree with me, you at least have to contradict me.

[/ QUOTE ]

Umm...I did contradict you. You said "but the most credit goes for doing it entirely with arithmetic", which I took to mean that you thought Doug was using something other than arithmetic. My point was that Doug did do it entirely with arithmetic (okay, and maybe a subconcious application of simple algebra from the distant past ).

[ QUOTE ]
[ QUOTE ]
Using a specific simple concrete example with round numbers (100 miles distance, for example) to solve a general problem is the best way to quickly compute.

[/ QUOTE ]You can't solve a general problem with specific numbers, only the specific problem.

[/ QUOTE ]

Okay, not the best choice of words on my part. The point I was making is that it's often easier to understand a general idea by working through a specific example that demonstrates the general idea in concrete terms. That's why you work through all those math problems in school rather than just study the theory. 