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Motion Problems:

Motion Problems:. By Dr. Marcia Tharp and Dr. Julia Arnold . Motion problems use the equation D = RT where D is the distance traveled , R is the rate when traveling and T is the time spent traveling .

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Motion Problems:

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  1. Motion Problems: By Dr. Marcia Tharp and Dr. Julia Arnold

  2. Motion problems use the equation D = RT where D is the distance traveled, R is the rate when traveling and T is the time spent traveling. It is helpful to use a D = RT grid when solving motion problems as shown in the following example.

  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.)

  4. Rate Time Distance Juan 60 x 60x Amal 72 x 72x We are looking for a time where Juan and Amal will be 30 miles apart. How do we represent the distance between the two men? 60x - 72x Or is it 72x - 60x. Which of these two would be positive? The correct equation is 72x - 60 x = 30

  5. The purpose of the grid is to find an algebraic name for each distance. Notice that the distance 30 miles does not appear in the grid because neither Juan nor Amal traveled 30 miles. Notice also that we could use x for each time since Juan and Amal were on the road for the same amount of time. We will need to work 30 miles into the equation as follows: 72x – 60x = 30 12x = 30 x = 2.5 hrs.

  6. Rate Time Distance Sherry Bob 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 know about Sherry and Bob. When finished go to the next slide to see how you did.

  7. Rate Time Distance Sherry Bob 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? Let x = the time it takes Bob to catch Sherry x + 1/2 5(x + 1/2) 5 7 x 7x Sherry started 30 minutes or 1/2 hour before Bob, so her time must reflect that amount. Rate is in miles per hour, time must be in hours so our units match.

  8. Sherry’s jogging distance Sherry Bob The problem is finished when Bob catches up to Sherry. Thus, their distances must equal each other. 5(x + 1/2) = 7x 5/2 = 2x 2[5/2] = 2 (2x) 5 = 4x 5x + 5/2 = 7x x = 5/4 = 1.25 hrs

  9. Now it’s time to see what You can do.

  10. Practice Problems: Motion • 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? • 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.

  11. 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?

  12. For worked out solutions click to next slide.

  13. 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.

  14. 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

  15. 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: 52x Tonya 39 mi 65x Freda Do you see the equation forming from our picture?

  16. 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? 52x Tonya 39 mi 65x Freda One way to look at it might be to say 52x + 39 = 65x Or another way might be to say 65x – 52x = 39 Both are correct and will give you the correct answer of 3 hrs.

  17. 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.

  18. 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: START 52x 65x Freda Tonya 39 miles Do you see the equation forming from this picture?

  19. 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. START 52x 65x Tonya 39 miles Freda 52x + 65x = 39 117x=39 X= 1/3 of an hour or 1/3 of 60 min = 20 minutes

  20. 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, lets Fill in the chart.

  21. 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? t 60t 2.2 - t 48(2.2 – t) 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. Distance is again computed by formula R*T = D

  22. 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? t 60t 2.2 - t 48(2.2 – t) What is the picture for this problem? 60t 48(2.2 – t) Dist 1st leg Dist 2nd leg Total dist. 120 miles Do you see the equation from the picture? 60t +48(2.2 – t) = 120

  23. 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? 60t 48(2.2 – t) Dist 1st leg Dist 2nd leg Total dist. 120 miles 60t +48(2.2 – t) = 120 60t + 105.6 – 48t = 120 12t = 14.4 t = 1.2 hours at 60mph

  24. 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.

  25. 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? 6x 6 5 5(x + 9) The problem is over when the drummer passes Dr. John at 6 PM, so how long has Dr. John been driving? How long has the drummer been driving? What is our picture? 6x Dr. John 5x + 45 Drummer Drummer passing Dr. John

  26. 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 + 45 X = 45 mph for Dr. John X + 9 = 54 mph for the drummer

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