Helicopters Over a Traffic Jam

Date: 09/22/98 at 20:29:50
From: CWS
Subject: Relative speeds

This is a math problem related to dealing with relative speeds:

A helicopter flies over a traffic jam with a velocity of 180 km/h. If 
it flies in the same direction the cars move, it needs 2 minutes to
overfly the cars. Flying back against the direction in which the cars 
are moving takes 80 seconds. How long is the traffic jam and with what
speed do the cars move?

Date: 09/23/98 at 09:21:55
From: Doctor Peterson
Subject: Re: Relative speeds

Hi, Martin.

Probably the hardest part about this problem is working with the 
units, since you have km/h and minutes and seconds all mixed together. 
Let's start by getting everything in terms of meters per second:

       km   1000 m    1 hr     1 min       m
   180 -- * ------ * ------ * ------ = 50 ---
       hr    1 km    60 min   60 sec      sec

Now suppose the traffic is moving at V m/sec.

If the helicopter flies in the same direction as the cars, it is moving 
at (50 - V) relative to the cars. It takes 120 sec to travel D (the 
length of the line), so:

   120 * (50 - V) = D

If it flies back, it is moving at (50 + V) relative to the cars. This 
takes 80 seconds, so:

   80 * (50 + V) = D

See if you can solve those two equations. One way is just to set the 
two D's equal to each other and solve for V, then find D from either 
of the two equations. You'll probably want to change V back into km/hr 
when you have it.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   

Date: 09/23/98 at 14:32:14
From: CWS
Subject: Re: Relative speeds

Hi,

First of all, thanks for answering my question. Second, I'd like to ask 
if I solved the equation correctly:

   D = 120*(50-V)
   D = 80*(50+V)

Substituting:

   120*(50-V) = 80*(50+V)

Thus:

   V = 10 km/h

And:

   D = 120*(50-10) = 340 meters

A: The traffic jam is 340 meters long and it moves at 10km/h.

Is this right?

Sincerely,
Martin

Date: 09/23/98 at 16:48:40
From: Doctor Peterson
Subject: Re: Moving-problem (puzzle?)

Hi, Martin. You're almost right! But you missed my reminder to change 
V back to km/h when you are done. It's 10 m/s. In calculating D you 
must have done something wrong, because I get 4800 m immediately. 
(Note that you do use the 10 in the formula for D, but you must 
remember again that the variable is in meters.)

When you get your final answers in a problem like this, it would be 
good to check them not only in the equations you wrote, but also in 
the actual problem as originally stated, using the actual units of 
your answer. How long will a helicopter going 180 km/h over 4.8 km of 
traffic moving 36 km/h take in each direction? Then you don't have to 
ask me if you are right. (And I don't have to check my own answers for 
you, which could be wrong.)

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   

Date: 09/24/98 at 21:07:29
From: CWS
Subject: Moving-Problem

Greetings,

I'd just like to say that I'm thankful to you for helping me with this
puzzle. It really helped me a lot.

Sincerely,
CWS