Distance is a measurement that answers
“how far?”
Time measures the length of time to
travel that distance and answers “how long?”
Rate is a measurement that answers “how
fast?”
The three are related this way:
D = R * T
D, distance, is R, rate, multiplied by
T, time. Distance, rate and time problems require attention to
units involved and deciding which one D, R, or T is the same between
moving objects.
First, if we are traveling 60 mph for
3 hours, how far did we travel. How far means distance, so
we have D = 60 mph * 3 hours.
Before you multiply let's write this another way:
Notice how the unit hour in the denominator cancels the hrs next
to the 3. This is very important, the units of a dimension must
match, if not, unit conversions must be done first. Time units must
be the same, distance units must be the same, and rate units must be
the same. See further down.
Anyway, the answer is 180 miles.
Let's state this another way, suppose
we were traveling 60 mph and traveled 180 miles. How long were we
on the road?
Here we have D, distance, again. And
we have the rate, R. We want to know how long?
Well, D = R * T, so
180 miles = 60mph * T
If we divide 180 by 60 (units match)
we get 3. T = 3 hours.
One last variation, if we traveled 180 miles
in 3 hours, how fast were we traveling?
D = R * T; 180 miles
= R * 3 hours if we divide by 3 we get 180/3 = 60 mph
Unit issues:
Suppose we traveled for 8
hours at a rate of 60 feet per second. How far did we travel?
Well time is in hours, rate is in
seconds. Time units must be the same. So first we need to convert
one to the other. Let's convert the 8 hrs to seconds. Well we have
60 sec in each minute, and 60 minutes in an hour, so 8 hours would be
8 * 60 * 60 secs = 28800 seconds. Now we can proceed and get
D = R * T ; D = 60
ft/sec * 28800 secs = 1728000 feet.
Another example:
A ball rolled 50 yards at 4 feet per
second. How long did the ball roll?
Here we have yards and feet; they must
be the same, so lets change the yards to feet. We have 3 feet in
each yard and 50 yards so we must have 3 * 50 = 150 feet. Now, we're
ready,
D = R * T; 150 feet
= 4 ft/sec * T
T must be 150/4 = 37.5 seconds
This next example is a bit more advanced
and a diagram can be helpful as shown.
John started to walk around a
track that is 1500 feet around. He is walking at 3 mph. Sarah
starts 5 minutes later at a light jog of 6 mph. How long will it
take Sarah to catch up to John?
3 mph is about 4 feet per
second, 6 mph is about 9 feet per second.
John is 5 minutes ahead,
which means he's covered 4 * 5 * 60 = 1200 feet already. In the
diagram below you can see John's head start. Sarah then starts to run, and
John is still walking (blue lines), eventually Sarah catches up to John (at the
right end of this diagram.)
Once Sarah starts to run, when she
overtakes John, both will have traveled a distance D
(shown above) fps = feet per second
The distance that Sarah will have run
is D = 9fps * T
This distance is the same as what John has covered
which is
the initial 1200 feet plus 4fps * T.
the distance Sarah runs =
distance John walks
9fps * T = 1200ft
+ 4fps * T
(units are consistent: fps) We can
subtract the 4fps T from the 9 fpsT to get 5fpsT
9fps * T -
4fps*T = 1200 ft
5 fpsT
= 1200 feet
1200 / 5 = 240 seconds = 240/60 = 4
minutes.
After Sarah catches up to John, how
long will it take foe her to up with him again?
This time they are shoulder to
shoulder. Sarah will run around the track and reach this same
location before catching up to John. This is the same as before but
it's as if John had a head start of one complete track length. As
before we have
John has run
1500 ft + 4fpsT
and Sarah
9fps T
1500 ft
+ 4fps*T = 9fps*T
1500 feet = 9fps*T - 4fps*T
T = 1500 / 5
= 300 secs = 300/60 = 5 minutes.