



Now a bit
of info about flying saucers (and airplanes too). 


The speed
that an airplane travels over the ground 


is a
combination of two things. 





One is
the speed that it moves through the air 


(called
airspeed). 


The other
is the speed that the air is moving 


(called
wind). 





Human
pilots try to fly routes where the wind is blowing 


in the
same direction that they are flying. 


That lets
them move faster over the ground. 


They get
to the destination faster and use less fuel. 


(Alien's
probably do the same thing, 


but I've
never had the chance to ask.) 





Example: 





The wind
is blowing from the north to the south 


at 10
miles per hour. 





Two
airplanes leave the same airport at the same time 


AND
TRAVEL AT THE SAME AIRSPEED. 





One
travels north (into the wind) 


The other
travels south (with the wind) 





At some
time later, the one that was traveling north 


is 200
miles from the airport. 


The one
traveling south is 235 miles from the airport. 





What is
the airspeed of the two planes?






The
problems are starting to get more involved. 


We are
going to use: 





D for distance 


R for rate (the
speed the plane flies OVER THE GROUND) 


T for time. 





We also
need to tell the two planes apart. 


To do
this, we call the plane going north plane 1. 


Then we
can call the distance it went D_{1} 


and the
speed it went over the ground R_{1} 


and the
time it traveled T_{1}. 





We can
also call the plane that went south, plane 2. 


Plane 2
can use D_{2} , R_{2} , and T_{2} for it's
distance, rate and time. 





So what
of these things do we know? 





D_{1}
= 200 miles 
D_{2}
= 235 miles 
R_{1}
= airspeed  10 
R_{2}
= airspeed + 10 
T_{1}
= unknown, but equal to T_{2} 
T_{2}
= unknown, but equal to T_{1} 






Here's
what we do. 


We have
too many things that are unknown. 


We need
to get rid of a few of them. 





We don't
know either of the times, 


but we
know that they are the same. 


That
means we can write them both as T. 





The
ground speeds of the two planes 


are
different because of the wind. 


But the
airspeeds are the same. 


If we
call the airspeed r, then 


R_{1}
= r 10 and R_{2} = r + 10. 





The DRT
equations for the two planes are: 





D_{1} =
R_{1} x T
and D_{2} = R_{2} x T 





Substituting
what we know, we have ... 





200 = (r  10) x
T and 235 = (r + 10)
x T 





Now we
have two equations with 2 unknowns. 


There are
a lot of ways that we can solve these. 





In the
way we are going to use, 


the first
step is to turn them each around to be: 





T
= "Stuff"












Here's
the tricky part. 


Since
"T" is the same in both equations, 


we can
say ... 











and
actually, we don't even need the "T" ... 











Now we
solve for "r" (which is the airspeed) 





Multiply
both sides by (r  10)(r + 10) to get rid of the denominators, 


and
simplify ... 











So
airspeed is about 124.29 miles per hour. 


Hmm, I
don't think that these planes are F16's. 





OK, now
we look back at the original problem, 


and make
sure we have actually answered 


what they
were looking for. 





We have. 





They
MIGHT have asked for something else, 


like the
Time they have been flying, 


or the
groundspeed of one or both planes. 





They
didn't. 





copyright 2005 Bruce Kirkpatrick 
