Section 3.4 Motion Practice

3.4 Motion PracticeStep-by-Step

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Instructions

Each problem in this presentation is worked out in several steps.

After reading a problem, try to do it on your own.

Check each step.

If you find that you have made a misstep, rework the problem from that stage.

This presentation contains six problems.

Problem 1

Train A leaves a station traveling at 40 mph. Eight hours later, train B leaves the same station traveling in the same direction at 60 mph. How long does it take for train B to catch up to train A?

Problem 1 (step 1a)

What are the two distances we are working with?

What is the relationship between the two distances?

Problem 1 (step 1b)

What are the two distances we are working with? The distances traveled by the two trains.

What is the relationship between the two distances? They are equal.

Problem 1 (step 2a)

Identify the rate and time of each moving object and use these to find expressions for distance.

Problem 1 (step 2b)

Rate (miles per hour)

Time (hours)

Distance (miles)

Train A 40 t+8

Train B 60 t

Problem 1 (step 2c)

Time (hours)

Distance (miles)

Train A 40 t+8 40(t+8)

Train B 60 t 60t

Problem 1 (step 3a)

Use the expressions for the distances and the relationship between the distances to create an algebraic equation.

Problem 1 (step 3b)

Time (hours)

Distance (miles)

Train A 40 t+8 40(t+8)

Train B 60 t 60t

40(t+8) = 60t

Problem 1 (step 4a)

Solve the equation you have found.

40(t+8) = 60t

Problem 1 (step 4b)

40(t+8) = 60t

40t + 320 = 60t

320 = 60t – 40t

320 = 20t

16 = t

Problem 1 (step 5a)

Answer the question asked. Include units.

Problem 1 (step 5b)

16 hours

Problem 2

A Miata and a Hummer enter an expressway at the same time and place and head the same direction. The Miata travels 77 miles per hour and the Hummer travels at 63 miles per hour. In how many hours will they be 21 miles apart?

Problem 2 (step 1a)

Problem 2 (step 1b)

What are the two distances we are working with? The distance traveled by the Miata and the distance traveled by the Hummer.

What is the relationship between the two distances? The Miata distance minus the Hummer distance is 21 miles.

Problem 2 (step 2a)

Problem 2 (step 2b)

Time (hours)

Distance (miles)

Miata 77 t

Hummer 63 t

Problem 2 (step 2c)

Time (hours)

Distance (miles)

Miata 77 t 77t

Hummer 63 t 63t

Problem 2 (step 3a)

Problem 2 (step 3b)

Time (hours)

Distance (miles)

Miata 77 t 77t

Hummer 63 t 63t

77t – 63t = 21

Problem 2 (step 4a)

77t – 63t = 21

Problem 2 (step 4b)

77t – 63t = 21

14t = 21

t = 21/14

t = 1.5

Problem 2 (step 5a)

Problem 2 (step 5b)

1.5 hours

Problem 3

When Keith works out, he first warms up by walking for 30 minutes at one speed. He then jogs for an hour at twice his walking speed. The total distance he travels is 10 miles. How fast does he walk?

Problem 3 (step 1b)

What are the two distances we are working with? The distance that Keith walks and the distance that he runs.

What is the relationship between the two distances? The sum of these distances is 10 miles.

Problem 3 (step 2a)

Problem 3 (step 2b)

Time (hours)

Distance (miles)

Walking r 0.5

Jogging 2r 1

Problem 3 (step 2c)

Time (hours)

Distance (miles)

Walking r 0.5 0.5r

Jogging 2r 1 2r

Problem 3 (step 3a)

Problem 3 (step 3b)

Time (hours)

Distance (miles)

Walking r 0.5 0.5r

Jogging 2r 1 2r

0.5r + 2r = 10

Problem 3 (step 4a)

0.5r + 2r = 10

Problem 3 (step 4b)

0.5r + 2r = 10

2.5 r = 10

r = 10 / 2.5

Problem 3 (step 5a)

Problem 3 (step 5b)

4 miles per hour

Problem 4

Raul drove roundtrip on an overnight visit to his grandparents. His average speed was 44 miles per hour on the way there and 52 miles per hour on the way back. He drove for a total of 12 hours during this trip.

How long did it take him to drive to his grandparents house?

How far away do his grandparents live?

Problem 4 (step 1a)

How far away do his grandparents live?What are the two distances we are working with?What is the relationship between the two

distances?

Problem 4 (step 1b)

How far away do his grandparents live?What are the two distances we are working with?

The distance to Raul’s grandparents’ house and the distance from Raul’s grandparents’ house.

What is the relationship between the two distances? They are equal.

Problem 4 (step 2a)

Problem 4 (step 2b)Raul drove roundtrip on an overnight visit to his grandparents. His

average speed was 44 miles per hour on the way there and 52 miles per hour on the way back. He drove for a total of 12 hours during this trip.

How long did it take him to drive to his grandparents house?How far away do his grandparents live?Identify the rate and time of each moving object and use these to find

expressions for distance.

Time (hours)

Distance (miles)

To 44 t

From 52 12 - t

Problem 4 (step 2c)Raul drove roundtrip on an overnight visit to his grandparents. His

average speed was 44 miles per hour on the way there and 52 miles per hour on the way back. He drove for a total of 12 hours during this trip.

How long did it take him to drive to his grandparents house?How far away do his grandparents live?Identify the rate and time of each moving object and use these to find

expressions for distance.

Time (hours)

Distance (miles)

To 44 t 44t

From 52 12 - t 52(12 – t)

Problem 4 (step 3a)

Problem 4 (step 3b)

Time (hours)

Distance (miles)

To 44 t 44t

From 52 12 - t 52(12 – t)

44t = 52(12 – t)

Problem 4 (step 4a)

44t = 52(12 – t)

Problem 4 (step 4b)

44t = 52(12 – t)

44t = 624 – 52t

44t + 52t = 624

96 t = 624

t = 624/96

t = 6.5

Problem 4 (step 5a)

Problem 4 (step 5b)

Answer the question asked. Include units.How long did it take him to drive to his

grandparents house? 6.5 hoursHow far away do his grandparents live?

44(6.5) = 286 286 miles

Problem 5

A passenger train and a freight train pass each other on parallel tracks heading in opposite directions. The freight train is traveling 12 miles per hour slower than the passenger train. After 5 hours are they 540 miles apart. How fast is each train traveling?

Problem 5 (step 1a)

Problem 5 (step 1b)A passenger train and a freight train pass

each other on parallel tracks heading in opposite directions. The freight train is traveling 12 miles per hour slower than the passenger train. After 5 hours are they 540 miles apart. How fast is each train traveling?

What are the two distances we are working with? The distance traveled by the Passenger Train and the distance traveled by the Freight Train.

What is the relationship between the two distances? The sum of the distances is 540 miles.

Problem 5 (step 2a)

Problem 5 (step 2b)

Time (hours)

Distance (miles)

Passenger r 5

Freight r-12 5

Problem 5 (step 2c)

Time (hours)

Distance (miles)

Passenger r 5 5r

Freight r – 12 5 5(r – 12)

Problem 5 (step 3a)

Problem 5 (step 3b)

Time (hours)

Distance (miles)

Passenger r 5 5r

Freight r – 12 5 5(r – 12)

5r + 5(r – 12) = 540

Problem 5 (step 4a)

5r + 5(r – 12) = 540

Problem 5 (step 4b)

5r + 5(r – 12) = 540

5r + 5r – 60 = 540

10r = 540 + 60

10r = 600

r = 600/10

r = 60

Problem 5 (step 5a)

Problem 5 (step 5b)

Answer the question asked. Include units.Passenger Train (r): 60 mph

Freight Train (r – 12): 48 mph

Problem 6

Janelle drove her friend to the train station and they arrived at 12:00 noon, 15 minutes too late for the train. Janelle decided she should drive her friend to the next major stop. The train travels at 50 miles per hour and is scheduled to arrive at its next major stop at 1:45 pm. How fast does Janelle need to drive to meet it when it arrives?

Problem 6 (step 1a)

Problem 6 (step 1b)

What are the two distances we are working with? The distance traveled by the train and the distance traveled by Janelle.

What is the relationship between the two distances? They are equal

Problem 6 (step 2a)

Janelle drove her friend to the train station and they arrived at 12:00 noon, 15 minutes too late for the train. Janelle decided she should drive her friend to the next major stop. The train travels at 50 miles per hour and is scheduled to arrive at its next major stop at 1:45 pm. How fast does Janelle need to drive?

Problem 6 (step 2b)

Time (hours)

Distance (miles)

Train 50 2

Janelle r 1.75

Problem 6 (step 2c)Janelle drove her friend to the train station and they arrived at 12:00

noon, 15 minutes too late for the train. Janelle decided she should drive her friend to the next major stop. The train travels at 50 miles per hour and is scheduled to arrive at its next major stop at 1:45 pm. How fast does Janelle need to drive to meet it when it arrives?

Time (hours)

Distance (miles)

Train 50 2 50(2)

Janelle r 1.75 1.75r

Problem 6 (step 3a)

Problem 6 (step 3b)

Time (hours)

Distance (miles)

Train 50 2 50(2)

Janelle r 1.75 1.75r

50(2) = 1.5r

Problem 6 (step 4a)

50(1.75) = 1.5r

Problem 6 (step 4b)

50(1.75) = 1.75r

100 = 1.75 r

100/1.75 = r

Problem 6 (step 5a)

Problem 6 (step 5b)

miles per hour

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