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Applied Problems: Motion
By Dr. Marcia L.Tharp
Dr. Julia Arnold
Applied Problems: Motion
Motion problems use the equation
D = RT
where D is the distance traveled.R is the rate of travel.T is the time spent traveling.
Applied Problems: MotionMotion Problem 1
Let’s suppose that you drive your car 120 miles in 2 hours than we can find the rate you traveled by using the distance equation.
D = RT
60 = R
Since 120 is a distance we let D = 120Since the time you traveled is 2 hours we let T= 2.
Placing this in the equation D=RT we have 120 = R 2
120 = R 2
2 2
So your rate is 60 miles per hour.
Motion Problem 2 Pat has been caught speeding by
the airplane patrol. He is doing 85miles per hour. A police car who is 30 miles behind him can do 100 miles per hour to catch him. How long will it take the police car to catch Pat if they continue in the same direction at this speed?
Print this page.
It is helpful to use a D = RT grid when solving motion problems as shown in the following example. The purpose of the grid is to find an algebraic name for each distance.
D=RT GridTo use this grid fill in the rate and time for each driver.We will let x = the time it takes for the police car to catch Pat. Then multiply the rate by the time to get the distance traveled.
Rate Time Distance
Pat
PoliceCar
D=RT Grid
Rate Time Distance
Pat 85
PoliceCar
100
D=RT Grid
Rate Time Distance
Pat 85 X
Police Car
100 X
D=RT GridRate Time Distance
Pat 85 X 85x
Police Car
100 X 100x
Notice that neither the policeman or Pat are represented by the 30 miles. We will use this later.
Next Draw A Picture
Picture 30 miles Pat’s
distance
Police Car’s distance
Draw A Picture
Picture 30 miles Pat’s
distance
Police Car’s distance
Use the picture to write an equation.
30 miles Pat’s distance
Police Car’s distance
Police Car’s = 30 miles + Pat’s Distance Distance
Now fill in the equation using our chart.
30 miles Pat’s distance
Police Car’s distance
Police Car’s = 30 miles + Pat’s Distance Distance
100x = 30 + 85xRate Tim
eDistance
Pat 85 X 85x
Police Car
100
X 100x
Next solve this equation.
100x = 30 + 85x
So it took the police car 2 hours to catch Pat.
100x – 85x = 30 + 85x – 85x 15x = 30
15x = 30 15 15
x = 2 hours
Subtract 85 x from both sides.
Divide by 15 on each side.
Problem 3 Juan and Amal leave DC at the
same time headed south on I-95. If Juan averages 60 mph and Amal averages 72 mph how long will it take them to be 30 miles apart?
(Now would be a good time for a guess. Write yours down and try it in this table.)
Rate Time Distance
Juan 60 x 60x
Amal 72 x 72x
72x x72Amal
60x x60Juan
DistanceTimeRate
We are looking for a time where Juan and Amal will be30 miles apart. How do we represent the distance betweenthe two men?
60x - 72x
Or is it 72x - 60x. Which of these two would be positive?
The correct equation is 72x - 60 x = 30
Sherry and Bob like to jog in the park. Sherry can jog at 5 mph, while Bob can jog at 7 mph.If Sherry starts 30 minutes ahead of Bob, how long will it take Bob to catch up to Sherry?
Draw a grid like the one below and fill in what you knowabout Sherry and Bob.
Bob
Sherry
DistanceTimeRate
When finished go to the next slide to see how you did.
Sherry and Bob like to jog in the park. Sherry can jog at 5 mph, while Bob can jog at 7 mph.If Sherry starts 30 minutes ahead of Bob, how long will it take Bob to catch up to Sherry?
Bob
Sherry
DistanceTimeRate
5
7 x
Let x = the time it takes Bob to catch Sherry
x + 1/2 5(x + 1/2)
7x
Sherry started 30 minutes or 1/2 hour before Bob, so her timemust reflect that amount. Rate is in miles per hour, time mustbe in hours so our units match.
Sherry Sherry’s jogging distance
Bob
The problem is finished when Bob catches up to Sherry.
Thus, their distances must equal each other.
5(x + 1/2) = 7x
5x + 5/2 = 7x
5/2 = 2x2[5/2] = 2 (2x)5 = 4x
x = 5/4 = 1.25 hrs
Now it’s time to see what You can do.
Practice Problems: Motion1. Tonya and Freda drive away from Norfolk on the same road in the same direction. If Tonya is averaging 52 mph and Freda is averaging 65 mph, how long will it take for them to be 39 miles apart?
2. Tonya and Freda drive away from Norfolk on the same road in opposite directions. If Tonya is averaging 52 mph and Freda is averaging 65 mph, how long will it take for them to be 39 miles apart? Round your answer to the nearest minute.
3. Bernadette drove 120 miles. The first part of the trip she averaged 60 mph, but on the second part of the trip she ran into some congestion and averaged 48 mph. If the total driving time was 2.2 hours, how much time did she spend at 60 mph?
4. Dr. John left New Orleans at 12 noon. His drummer left at 1:00 traveling 9 mph faster. If the drummer passed Dr. John at 6:00, what was the average speed of each?
Motion problem solutions:1. 3 hours2. 1/3 hour = 20 minutes3. 1.2 hours at 60 mph4. 45 mph for Dr. John and 54 mph for the drummer
For worked out solutionsclick to next slide.
1. Tonya and Freda drive away from Norfolk on the same road in the same direction. If Tonya is averaging 52 mph and Freda is averaging 65 mph, how long will it take for them to be 39 miles apart? Let x = time it takes for them to be 39 miles apart. Construct a table to put your information in.
Rate Time Distance
Tonya 52 x 52x
Freda 65 x 65x
Tonya and Freda drive away from Norfolk on the same road in the same direction. If Tonya is averaging 52 mph and Freda is averaging 65 mph, how long will it take for them to be 39 miles apart?
We let x stand for the time they have been driving which would be the same in this case. How far they have gone (distance) is written in the chart by using the known information and the formula R*T=D
Rate Time Distance
Tonya 52 x 52x
Freda 65 x 65x
Tonya and Freda drive away from Norfolk on the same road in the same direction. If Tonya is averaging 52 mph and Freda is averaging 65 mph, how long will it take for them to be 39 miles apart?
Tonya and Freda are headed in the same direction so we can picture their distances as this:
Tonya
Freda39 mi
52x
65x
Do you see the equation forming from our picture?
Rate Time Distance
Tonya 52 x 52x
Freda 65 x 65x
Tonya and Freda drive away from Norfolk on the same road in the same direction. If Tonya is averaging 52 mph and Freda is averaging 65 mph, how long will it take for them to be 39 miles apart?
Tonya
Freda39 mi
52x
65x
One way to look at it might be to say52x + 39 = 65xOr another way might be to say 65x – 52x = 39Both are correct and will give you the correct answer of 3 hrs.
2. Tonya and Freda drive away from Norfolk on the same road in opposite directions. If Tonya is averaging 52 mph and Freda is averaging 65 mph, how long will it take for them to be 39 miles apart? Round your answer to the nearest minute.Now they are going in opposite directions, but we will begin the same way by constructing our table.
Rate Time Distance
Tonya 52 x 52x
Freda 65 x 65x
Tonya and Freda drive away from Norfolk on the same road in opposite directions. If Tonya is averaging 52 mph and Freda is averaging 65 mph, how long will it take for them to be 39 miles apart? Round your answer to the nearest minute.
As you can see the table is the same, so only the picture of the event must change. Now it looks like this:
Tonya Freda52x 65x
Do you see the equation forming from this picture?
START
39 miles
Rate Time Distance
Tonya 52 x 52x
Freda 65 x 65x
Tonya and Freda drive away from Norfolk on the same road in opposite directions. If Tonya is averaging 52 mph and Freda is averaging 65 mph, how long will it take for them to be 39 miles apart? Round your answer to the nearest minute.
TonyaFreda
52x 65x
52x + 65x = 39117x=39X= 1/3 of an hour or 1/3 of 60 min = 20 minutes
START
39 miles
3. Bernadette drove 120 miles. The first part of the trip she averaged 60 mph, but on the second part of the trip she ran into some congestion and averaged 48 mph. If the total driving time was 2.2 hours, how much time did she spend at 60 mph? The question here deals with time. Again, letsFill in the chart.
Rate Time Distance
1st leg 60
2nd leg 48
Bernadette drove 120 miles. The first part of the trip she averaged 60 mph, but on the second part of the trip she ran into some congestion and averaged 48 mph. If the total driving time was 2.2 hours, how much time did she spend at 60 mph?
This time totals are given for both time traveled and distance traveled. Thus totals don’t belong in the chart. Bernadette did not travel 2.2 hours at 60 mph nor did she travel 2.2 hours at 48 mph. She traveled 2.2 hours total at both of those speeds. So how do we write this in the chart. Click to observe.
t
2.2 - t
60t
48(2.2 – t)
Distance is again computed by formula R*T = D
Rate Time Distance
1st leg 60
2nd leg 48
Bernadette drove 120 miles. The first part of the trip she averaged 60 mph, but on the second part of the trip she ran into some congestion and averaged 48 mph. If the total driving time was 2.2 hours, how much time did she spend at 60 mph?
What is the picture for this problem?
t
2.2 - t
60t
48(2.2 – t)
Do you see the equation from the picture?
Dist 1st leg Dist 2nd leg
60t 48(2.2 – t)
Total dist. 120 miles
60t +48(2.2 – t) = 120
Bernadette drove 120 miles. The first part of the trip she averaged 60 mph, but on the second part of the trip she ran into some congestion and averaged 48 mph. If the total driving time was 2.2 hours, how much time did she spend at 60 mph?
Dist 1st leg Dist 2nd leg
60t 48(2.2 – t)
Total dist. 120 miles
60t +48(2.2 – t) = 12060t + 105.6 – 48t = 12012t = 14.4 t = 1.2 hours at 60mph
4. Dr. John left New Orleans at 12 noon. His drummer left at 1:00 traveling 9 mph faster. If the drummer passed Dr. John at 6:00, what was the average speed of each?
Now we don’t know the speed. We do know something about time. Let’s see how we can fill in the chart.
Rate Time Distance
John x
Drummer
X + 9
Dr. John left New Orleans at 12 noon. His drummer left at 1:00 traveling 9 mph faster. If the drummer passed Dr. John at 6:00, what was the average speed of each?
The problem is over when the drummer passes Dr. John at 6 PM, so how long has Dr. John been driving?
6
How long has the drummer been driving?
5
What is our picture?
Dr. John
DrummerDrummer passing Dr. John
6x
5(x + 9)
6x
5x + 45
Rate Time Distance
John x 6x
Drummer
X + 9 5(x + 9)
Dr. John left New Orleans at 12 noon. His drummer left at 1:00 traveling 9 mph faster. If the drummer passed Dr. John at 6:00, what was the average speed of each?
6
5
Dr. John
Drummer
What equation does the picture suggest?
The two distances are equal thus:6x = 5(x + 9)6x = 5x + 45X = 45 mph for Dr. JohnX + 9 = 54 mph for the drummer
Back to Word Problem Menu Back to Math 03 Modules
Now go on to the Geometry Lessons.