Motion Word Problems

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Word Problems - Motion
By Joe Joyner Math 04 Intermediate Algebra
Link to Practice Problems
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Introduction
In this module, youll continue to develop and
work with mathematical models.
When solving practical application problems, you
try to find a mathematical model for the problem.

A mathematical model does not necessarily have to
be complicated. It can be relatively simple.
This is usually the case when only one or two
variables are required to build a linear model.
Lets begin.
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Rate, Time, and Distance Problems
If an object such as an automobile or an airplane
travels at a constant, or uniform, rate of speed,
r ,
then the distance traveled by the object, d,
during a period of time, t
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Rate, Time, and Distance Problems
is given by the distance, rate, time formula
d rt.
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Rate, Time, and Distance Problems
Example 1
You ride your bike for 7 hours. If you travel
36.75 miles, what is your average speed?
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Rate, Time, and Distance Problems
Example 1
The quantities in this problem are

• distance (constant at 36.75 miles),

• time (constant at 7 hours),

• and rate, or speed (unknown variable).

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Rate, Time, and Distance Problems
Example 1
You can use a spreadsheet (Excel, for example) to
build a model for this problem.
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Rate, Time, and Distance Problems
Example 1
Explore
To access the spreadsheet, click the word
Explore.
Then explore with the rate to see if you can
solve the problem.
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Rate, Time, and Distance Problems
Example 1
Represent the variable rate with r .
You can use the distance, rate, time formula.
d rt
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Rate, Time, and Distance Problems
Example 1
But since you know the distance and time, and
wish to solve for rate, it would be helpful to
solve the equation for r first.
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Rate, Time, and Distance Problems
Example 1
is our mathematical model.
Some mathematical models can be easy!
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Rate, Time, and Distance Problems
Example 1
Now we can solve for the rate, r , by dividing
the distance by the time.
5.25 miles per hour
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Rate, Time, and Distance Problems
When you read a word problem that involves rate,
time, and distance, note whether the problem
situation involves

• motion in the same direction

• motion in opposite directions

• a round trip.

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Rate, Time, and Distance Problems
Example 2
Dan and Emily are truck drivers. Dan, averaging
55 miles per hour (mph), begins a 280-mile trip
from their companys Norfolk warehouse to
Charlotte, NC at 7 AM.
Emily sets out from the Charlotte warehouse at
8 AM on the same day as Dan and travels at
45 mph in the opposite direction as the route
taken by Dan.
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Rate, Time, and Distance Problems
Example 2
How many hours will Emily have been driving when
she and Dan pass each other?
How will you start to set up a model for solving
this problem?
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Rate, Time, and Distance Problems
Example 2
What is the variable that you must solve for?
time
Is the length of time traveled the same for Dan
and Emily when they pass each other?
No.
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Rate, Time, and Distance Problems
Example 2
Why is the time different for the two drivers?

• Dan started at 7 AM and
• Emily started at 8 AM.

• Dan averaged 55 mph and
• Emily averaged 45 mph.

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Rate, Time, and Distance Problems
Example 2
Let t represent the amount of time that Emily
travels until the trucks pass each other.
In terms of t , how long will Dan have been on
the road when the trucks pass each other?
t 1
One hour longer or ...
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Rate, Time, and Distance Problems
Example 2
You can use a spreadsheet to build a model for
this problem too.
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Rate, Time, and Distance Problems
Explore
Example 2
To access the spreadsheet, click the word
Explore.
Then explore with Emilys time to see if you can
solve the problem.
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Rate, Time, and Distance Problems
Example 2
The mathematical model for this problem is
Dans Distance Emilys Distance 280 miles
Dans rateDans time Emilys rateEmilys time
280
55(t1) 45t 280
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Rate, Time, and Distance Problems
Example 2
55(t1) 45t 280
55t55 45t 280
100t 55 280
100t 225
t 2.25 hours
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Rate, Time, and Distance Problems
Example 3
Jason and LeRoy are entered in a 26-mile marathon
race. Jasons average pace is 6 miles per hour
(mph) and LeRoys average pace is 8 mph. Both
runners start at the same time.
How far from the finish line will Jason be when
LeRoy crosses the finish line?
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Rate, Time, and Distance Problems
Example 3
What are the known constants?

• Jasons rate of 6 mph

• LeRoys rate of 8 mph

• Race distance of 26 miles

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Rate, Time, and Distance Problems
Example 3
What are the unknowns?

• The amount of time it takes LeRoy to finish the
  race

• The distance Jason has to run when LeRoy
  finishes

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Rate, Time, and Distance Problems
Example 3
Let LeRoys time be t .
What is the distance, rate, time, model for Leroy
in this problem?
8t 26
What is the solution for t ?
t 3.25 hours
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Rate, Time, and Distance Problems
Example 3
At the time that LeRoy crosses the finish line,
Jason has run for the same amount of time, t .
What is the model for how far Jason is from the
finish line at that time?
d 26 - 6(3.25)
d 6.5 miles
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Rate, Time, and Distance Problems
Do you think youve got the concept of solving
motion (rate, time distance) problems?
Look at the next slide.
If you want to try the interactive web site that
the slide came from, click on the word Explore to
go there.
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Explore
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Rate, Time, and Distance Problems
Hopefully, you are now ready to practice motion
problems for yourself. When you click the Go To
Practice Problems link below, your web browser
will open the practice problem set.
Go To Practice Problems