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)
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?
What are the two distances we are working with?
What is the relationship between the two distances?
Problem 1 (step 1b)
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?
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)
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?
Identify the rate and time of each moving object and use these to find expressions for distance.
Problem 1 (step 2b)
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?
Identify the rate and time of each moving object and use these to find expressions for distance.
Rate (miles per hour)
Time (hours)
Distance (miles)
Train A 40 t+8
Train B 60 t
Problem 1 (step 2c)
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?
Identify the rate and time of each moving object and use these to find expressions for distance.
Rate (miles per hour)
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)
Use the expressions for the distances and the relationship between the distances to create an algebraic equation.
Rate (miles per hour)
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)
Solve the equation you have found.
40(t+8) = 60t
40t + 320 = 60t
320 = 60t – 40t
320 = 20t
16 = t
Problem 1 (step 5a)
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?
Answer the question asked. Include units.
Problem 1 (step 5b)
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?
Answer the question asked. Include units.
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)
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?
What are the two distances we are working with?
What is the relationship between the two distances?
Problem 2 (step 1b)
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?
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)
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?
Identify the rate and time of each moving object and use these to find expressions for distance.
Problem 2 (step 2b)
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?
Identify the rate and time of each moving object and use these to find expressions for distance.
Rate (miles per hour)
Time (hours)
Distance (miles)
Miata 77 t
Hummer 63 t
Problem 2 (step 2c)
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?
Identify the rate and time of each moving object and use these to find expressions for distance.
Rate (miles per hour)
Time (hours)
Distance (miles)
Miata 77 t 77t
Hummer 63 t 63t
Problem 2 (step 3a)
Use the expressions for the distances and the relationship between the distances to create an algebraic equation.
Problem 2 (step 3b)
Use the expressions for the distances and the relationship between the distances to create an algebraic equation.
Rate (miles per hour)
Time (hours)
Distance (miles)
Miata 77 t 77t
Hummer 63 t 63t
77t – 63t = 21
Problem 2 (step 4a)
Solve the equation you have found.
77t – 63t = 21
Problem 2 (step 4b)
Solve the equation you have found.
77t – 63t = 21
14t = 21
t = 21/14
t = 1.5
Problem 2 (step 5a)
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?
Answer the question asked. Include units.
Problem 2 (step 5b)
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?
Answer the question asked. Include units.
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)
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?
What are the two distances we are working with?
What is the relationship between the two distances?
Problem 3 (step 1b)
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?
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)
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?
Identify the rate and time of each moving object and use these to find expressions for distance.
Problem 3 (step 2b)
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?
Identify the rate and time of each moving object and use these to find expressions for distance.
Rate (miles per hour)
Time (hours)
Distance (miles)
Walking r 0.5
Jogging 2r 1
Problem 3 (step 2c)
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?
Identify the rate and time of each moving object and use these to find expressions for distance.
Rate (miles per hour)
Time (hours)
Distance (miles)
Walking r 0.5 0.5r
Jogging 2r 1 2r
Problem 3 (step 3a)
Use the expressions for the distances and the relationship between the distances to create an algebraic equation.
Problem 3 (step 3b)
Use the expressions for the distances and the relationship between the distances to create an algebraic equation.
Rate (miles per hour)
Time (hours)
Distance (miles)
Walking r 0.5 0.5r
Jogging 2r 1 2r
0.5r + 2r = 10
Problem 3 (step 4a)
Solve the equation you have found.
0.5r + 2r = 10
Problem 3 (step 4b)
Solve the equation you have found.
0.5r + 2r = 10
2.5 r = 10
r = 10 / 2.5
r = 4
Problem 3 (step 5a)
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?
Answer the question asked. Include units.
Problem 3 (step 5b)
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?
Answer the question asked. Include units.
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)
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?What are the two distances we are working with?What is the relationship between the two
distances?
Problem 4 (step 1b)
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?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)
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.
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.
Rate (miles per hour)
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.
Rate (miles per hour)
Time (hours)
Distance (miles)
To 44 t 44t
From 52 12 - t 52(12 – t)
Problem 4 (step 3a)
Use the expressions for the distances and the relationship between the distances to create an algebraic equation.
Problem 4 (step 3b)
Use the expressions for the distances and the relationship between the distances to create an algebraic equation.
Rate (miles per hour)
Time (hours)
Distance (miles)
To 44 t 44t
From 52 12 - t 52(12 – t)
44t = 52(12 – t)
Problem 4 (step 4a)
Solve the equation you have found.
44t = 52(12 – t)
Problem 4 (step 4b)
Solve the equation you have found.
44t = 52(12 – t)
44t = 624 – 52t
44t + 52t = 624
96 t = 624
t = 624/96
t = 6.5
Problem 4 (step 5a)
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?
Answer the question asked. Include units.
Problem 4 (step 5b)
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.
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)
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?
What is the relationship between the two distances?
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)
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?
Identify the rate and time of each moving object and use these to find expressions for distance.
Problem 5 (step 2b)
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?
Identify the rate and time of each moving object and use these to find expressions for distance.
Rate (miles per hour)
Time (hours)
Distance (miles)
Passenger r 5
Freight r-12 5
Problem 5 (step 2c)
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?
Identify the rate and time of each moving object and use these to find expressions for distance.
Rate (miles per hour)
Time (hours)
Distance (miles)
Passenger r 5 5r
Freight r – 12 5 5(r – 12)
Problem 5 (step 3a)
Use the expressions for the distances and the relationship between the distances to create an algebraic equation.
Problem 5 (step 3b)
Use the expressions for the distances and the relationship between the distances to create an algebraic equation.
Rate (miles per hour)
Time (hours)
Distance (miles)
Passenger r 5 5r
Freight r – 12 5 5(r – 12)
5r + 5(r – 12) = 540
Problem 5 (step 4a)
Solve the equation you have found.
5r + 5(r – 12) = 540
Problem 5 (step 4b)
Solve the equation you have found.
5r + 5(r – 12) = 540
5r + 5r – 60 = 540
10r = 540 + 60
10r = 600
r = 600/10
r = 60
Problem 5 (step 5a)
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?
Answer the question asked. Include units.
Problem 5 (step 5b)
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?
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)
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?
What are the two distances we are working with?
What is the relationship between the two distances?
Problem 6 (step 1b)
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?
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?
Identify the rate and time of each moving object and use these to find expressions for distance.
Problem 6 (step 2b)
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?
Identify the rate and time of each moving object and use these to find expressions for distance.
Rate (miles per hour)
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?
Identify the rate and time of each moving object and use these to find expressions for distance.
Rate (miles per hour)
Time (hours)
Distance (miles)
Train 50 2 50(2)
Janelle r 1.75 1.75r
Problem 6 (step 3a)
Use the expressions for the distances and the relationship between the distances to create an algebraic equation.
Problem 6 (step 3b)
Use the expressions for the distances and the relationship between the distances to create an algebraic equation.
Rate (miles per hour)
Time (hours)
Distance (miles)
Train 50 2 50(2)
Janelle r 1.75 1.75r
50(2) = 1.5r
Problem 6 (step 4a)
Solve the equation you have found.
50(1.75) = 1.5r
Problem 6 (step 4b)
Solve the equation you have found.
50(1.75) = 1.75r
100 = 1.75 r
100/1.75 = r
r =
Problem 6 (step 5a)
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?
Answer the question asked. Include units.
Problem 6 (step 5b)
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?
Answer the question asked. Include units.
miles per hour