Velocity and Time

Date: 02/11/99 at 08:31:34
From: Jenny and Diana
Subject: Physics - velocity, displacement, and period

Hi Dr. Math. We are trying to figure the problem below, but we can't 
get the correct answer. The right answer is 37.5 km/hr but we keep 
coming up with the answer of 40 km/hr. Please help us.

A car averages 30 km/hr the first half of a trip, and 50 km/hr the 
second half of the trip. What is the car's average velocity for the 
trip?

Thank you so much for your help!

Sincerely,
Jenny and Diana

Date: 02/11/99 at 14:11:04
From: Doctor Micah
Subject: Re: Physics - velocity, displacement, and period

Hi Jenny and Diana,

Thanks for writing to Dr. Math. This is a very tricky problem. 
Depending on how you read the problem, you could argue that your answer 
(40) is the correct one.

The tricky part is where the problem states the car averages 30 km/hr
for the "first half" of the trip. What is unclear is, does it mean half 
the distance, or half the time? You are assuming that it means half the 
time, which is a pretty good assumption the way the problem is worded. 
So you calculate the average velocity like this:

total distance = (30 km/hr)*0.5t + (50 km/hr)*0.5t = (40 km/hr)*t
total time = t

                    (40 km/hr)*t
average velocity =  ------------- = 40 km/hr
                          t

What the problem obviously meant, though, since 40 is not the given
answer, is that the car traveled 30 km/hr for the first half of the
total distance. So:

total distance = d

                0.5d         0.5d
total time = ---------- + ---------
              30 km/hr     50 km/hr

I won't work out the details for you, but I think now that you see what 
the problem meant (though it wasn't at all clear), you can get the 
book's answer for yourself.  

If you have any more trouble, please feel free to write back to Dr. 
Math. Thanks for your question.

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