1. Solving harder physics problems
Some problems require multiple steps and solving for multiple unknown variables. For each unknown variable, an independent equation is needed. In this lesson, students apply a four-step problem solving method to more complex, multi-step physics problems.

2. Objectives
Convert quantities from one unit to another using appropriate conversion factors.
Use algebraic models to analyze and solve multi-step problems mathematically.
Use multiple independent equations to solve for multiple variables.
The lesson objectives clarify what students should know and be able to do.

3. Assessment
Two cars are initially separated by 1.0 km and traveling towards each other. One car travels at 20 miles per hour and the second car travels at 20 meters per second.
Convert all given quantities to metric units if needed.
How long does it take for the two cars to meet?
Question 1 addresses the first and second objectives. These formative assessments will appear again at the end of the lesson, along with the answers.

4. Assessment
A car travels a total distance of 2.0 kilometers. It travels the first half of the distance at a constant speed of 15 m/s. It travels the second half of the distance at a constant speed of 25 m/s. What is the average speed of the car?
This question addresses the third objective. These formative assessments will appear again at the end of the lesson, along with the answers.

5. Physics terms variable conversion factor position velocity
acceleration
All these physics terms have been introduced in previous lessons.

6. Equations
None of these equations is new. But in this lesson students will need to use more than one equation to solve a problem, and use their algebra skills to rearrange and combine expressions.

7. Solving harder physics problems
Two bicyclists approach each other on the same road. One has a speed of 5.0 m/s and the other has a speed of 8.0 m/s. They are 500 meters apart. How long will it be before they meet?
Students often struggle to know where to start when solving harder physics problems. Remind them of the four-step method from Chapter 2.

8. Assumptions you can make
In most problems you may assume the following:
Assume there is no friction, unless you are told otherwise.
Velocities are constant unless you know otherwise.
3. Initial position, initial time, and initial velocity are zero unless you know otherwise. zero, unless you know otherwise.
Point out to the students that in solving problems, some information is directly provided and some is implied in the problem statement.

9. How do you start? What are you asked for? What is given?
What is the relationship?
What is the solution?
These are the four steps. They will be used to analyze the problem.

10. Apply the four step method
Two bicyclists approach each other on the same road. One has a speed of 5.0 m/s and the other has a speed of 8.0 m/s. They are 500 meters apart. How long will it be before they meet?
Ask the students what the first step should be.

11. Find wanted and given values
Two bicyclists approach each other on the same road. One has a speed of 5.0 m/s and the other has a speed of 8.0 m/s. They are 500 meters apart. How long will it be before they meet?
What are you asked for?
Apply the four step method to the stated problem. Get students to answer the question before confirming the answer with the next slide.

12. Find wanted and given values
Two bicyclists approach each other on the same road. One has a speed of 5.0 m/s and the other has a speed of 8.0 m/s. They are 500 meters apart. How long will it be before they meet?
What are you asked for?
What is given?
time
time
Again, ask students to answer the question before confirming the answer with the next slide.

13. Find wanted and given values
Two bicyclists approach each other on the same road. One has a speed of 5.0 m/s and the other has a speed of 8.0 m/s. They are 500 meters apart. How long will it be before they meet?
What are you asked for?
What is given?
speed 1
speed 2
total distance
time
speed 1, speed 2, total distance

14. What is the relationship?
Two bicyclists approach each other on the same road. One has a speed of 5.0 m/s and the other has a speed of 8.0 m/s. They are 500 meters apart. How long will it be before they meet?
What are you asked for?
What is given?
What is the relationship?
time
speed 1, speed 2, total distance
Prompt the students: What equation do they know that relates these variables: time, speed, and distance?

15. What is the solution?
Two bicyclists approach each other on the same road. One has a speed of 5.0 m/s and the other has a speed of 8.0 m/s. They are 500 meters apart. How long will it be before they meet?
What are you asked for?
What is given?
What is the relationship?
What is the solution?
time
speed 1, speed 2, total distance
Point out that this relationship is a GENERAL relationship. It must be applied to EACH of the bicyclists to solve the problem.

16. What is the solution?
Two bicyclists approach each other on the same road. One has a speed of 5.0 m/s and the other has a speed of 8.0 m/s. They are 500 meters apart. How long will it be before they meet?
What are you asked for?
What is given?
What is the relationship?
What is the solution?
time
speed 1, speed 2, total distance
The next slide will explain why multiple equations are needed.
The solution will require more than one equation!

17. Multiple unknowns need multiple equations
In this problems, there are three unknowns:
Although we know the TOTAL distance, each bicyclist will travel a different portion of that distance. (d1, d2)
The wanted variable is time ( t )
To solve for three unknown quantities, three independent equations are needed.
This may be the first time students have worked with multiple equations to solve for multiple unknown values.

18. Strategy for solving the problem
Two bicyclists approach each other on the same road. One has a speed of 5.0 m/s and the other has a speed of 8.0 m/s. They are 500 meters apart. How long will it be before they meet?
Brainstorm with a partner: what three equations could be used to solve this problem?
Ask students to brainstorm with a partner about what equations could be used to solve the problem. Bring the group back together to solve the problem.

19. Solving the problem
Two bicyclists approach each other on the same road. One has a speed of 5.0 m/s and the other has a speed of 8.0 m/s. They are 500 meters apart. How long will it be before they meet?
General
relationship:
Point out that each bicyclist travels for the same time (t) and that their total distance traveled is 500 m. Ask students to identify the relationship among the variables distance, time, and velocity.

20. Solving the problem
Two bicyclists approach each other on the same road. One has a speed of 5.0 m/s and the other has a speed of 8.0 m/s. They are 500 meters apart. How long will it be before they meet?
General
relationship:
Ask the students how to write this general relationship to fit this problem.

21. Relationships in this problem:
Solving the problem
Two bicyclists approach each other on the same road. One has a speed of 5.0 m/s and the other has a speed of 8.0 m/s. They are 500 meters apart. How long will it be before they meet?
General
relationship:
Relationships in this problem:
Ask students to express this relationship for EACH bicyclist.

22. Relationships in this problem:
Solving the problem
Two bicyclists approach each other on the same road. One has a speed of 5.0 m/s and the other has a speed of 8.0 m/s. They are 500 meters apart. How long will it be before they meet?
General
relationship:
Relationships in this problem:
Point out that only the variable t is needed for time, since the two times are the same. Ask students to brainstorm a next step.

23. Solving the problem
Two bicyclists approach each other on the same road. One has a speed of 5.0 m/s and the other has a speed of 8.0 m/s. They are 500 meters apart. How long will it be before they meet?
Solve for d1 and d2:
Step through the problem solution. Start by solving the velocity equation for d1 and d2.

24. Solving the problem
Two bicyclists approach each other on the same road. One has a speed of 5.0 m/s and the other has a speed of 8.0 m/s. They are 500 meters apart. How long will it be before they meet?
Solve for d1 and d2:

25. Solving the problem
Two bicyclists approach each other on the same road. One has a speed of 5.0 m/s and the other has a speed of 8.0 m/s. They are 500 meters apart. How long will it be before they meet?
Solve for d1 and d2:
Substitute:
Substitute these expressions into the equation for total distance traveled. The substitution is shown on the next slide.

26. Solving the problem
Two bicyclists approach each other on the same road. One has a speed of 5.0 m/s and the other has a speed of 8.0 m/s. They are 500 meters apart. How long will it be before they meet?
Solve for d1 and d2:
Substitute:

27. Solve for wanted variable:
Solving the problem
Two bicyclists approach each other on the same road. One has a speed of 5.0 m/s and the other has a speed of 8.0 m/s. They are 500 meters apart. How long will it be before they meet?
Solve for d1 and d2:
Substitute:
Solve for wanted variable:
Solve this expression for the wanted variable. Emphasize that students should solve for the wanted variable symbolically first. What is the first step?

28. Solve for wanted variable:
Solving the problem
Two bicyclists approach each other on the same road. One has a speed of 5.0 m/s and the other has a speed of 8.0 m/s. They are 500 meters apart. How long will it be before they meet?
Solve for d1 and d2:
Substitute:
Solve for wanted variable:
Step through the problem solution. The first step is to factor out the time, t.

29. Another approach
Two bicyclists approach each other on the same road. One has a speed of 5.0 m/s and the other has a speed of 8.0 m/s. They are 500 meters apart. How long will it be before they meet?
Learning to solve tough problems using multiple equations is a very useful skill.
For this particular problem there is an easier way: let the red bike be your reference frame!

30. Reference frame: red bike
Two bicyclists approach each other on the same road. One has a speed of 5.0 m/s and the other has a speed of 8.0 m/s. They are 500 meters apart. How long will it be before they meet?
asked:
given:
relationship:
solution:
time
total distance = 500 m, speed of blue bike = ?
Ask the students for the speed of the blue bike in the red bike reference frame.

31. Reference frame: red bike
Two bicyclists approach each other on the same road. One has a speed of 5.0 m/s and the other has a speed of 8.0 m/s. They are 500 meters apart. How long will it be before they meet?
asked:
given:
relationship:
solution:
time
total distance = 500 m, speed of blue bike = 13 m/s
Tell the students: “Learning how to switch reference frames is a useful skill than can help you simplify a complex problem.” In this case, only one equation was needed.

32. The importance of units
In solving problems, if units are NOT consistent you must convert. For example:
How long does it take a car traveling at 30 mph to travel across an intersection that is 18 m wide?
What is asked for?
Step through the problem solution.

33. The importance of units
In solving problems, if units are NOT consistent you must convert. For example:
How long does it take a car traveling at 30 mph to travel across an intersection that is 18 m wide?
What is asked for?
What is given?
time
Step through the problem solution. Ask for student responses.

34. The importance of units
In solving problems, if units are NOT consistent you must convert. For example:
How long does it take a car traveling at 30 mph to travel across an intersection that is 18 m wide?
What is asked for?
What is given?
What is the relationship?
time
distance, velocity
Step through the problem solution.

35. The importance of units
In solving problems, if units are NOT consistent you must convert. For example:
How long does it take a car traveling at 30 mph to travel across an intersection that is 18 m wide?
What is asked for?
What is given?
What is the relationship?
What is the solution?
time
distance, velocity
Step through the problem solution.

36. The importance of units
In solving problems, if units are NOT consistent you must convert. For example:
How long does it take a car traveling at 30 mph to travel across an intersection that is 18 m wide?
What is asked for?
What is given?
What is the relationship?
What is the solution?
time
distance, velocity
What’s wrong here?
Allow students time to recognize the problem.

37. Covert to consistent units
How long does it take a car traveling at 30 mph to travel across an intersection that is 18 m wide?
Convert this velocity to metric units!

38. Solve
In solving problems, if units are NOT consistent you must convert. For example:
How long does it take a car traveling at 30 mph to travel across an intersection that is 18 m wide?
What is asked for?
What is given?
What is the relationship?
What is the solution?
time
distance, velocity
The correct answer is 1.3 m/s, NOT 18/30 = 0.6 seconds. Ignoring units leads to wrong answers.

39. Assessment
Two cars are initially separated by 1.0 km and traveling towards each other. One car travels at 20 miles per hour and the second car travels at 20 m/s.
Convert all given quantities to metric units if needed.
How long does it take for the two cars to meet?
The answer to part a appears on the next slide.

40. Assessment
Two cars are initially separated by 1.0 km and traveling towards each other. One car travels at 20 miles per hour and the second car travels at 20 m/s.
Convert all given quantities to metric units if needed.
How long does it take for the two cars to meet?
The answer to part b appears on the next slide.

41. Assessment
Two cars are initially separated by 1.0 km and traveling towards each other. One car travels at 20 miles per hour and the second car travels at 20 m/s.
How long does it take for the two cars to meet?

42. Assessment
A car travels a total distance of 2.0 kilometers. It travels the first half of the distance at a constant speed of 15 m/s. It travels the second half of the distance at a constant speed of 25 m/s. What is the average speed of the car?
The solution appears on the next two slides.

43. Assessment
A car travels a total distance of 2.0 kilometers. It travels the first half of the distance at a constant speed of 15 m/s. It travels the second half of the distance at a constant speed of 25 m/s. What is the average speed of the car?
wanted: average speed for the entire trip
given: dtotal = 2.0 km, v1 = 15 m/s, v2 = 25 m/s
relationships:
solution:
The solution appears on the next slide.

44. Assessment
A car travels a total distance of 2.0 kilometers. It travels the first half of the distance at a constant speed of 15 m/s. It travels the second half of the distance at a constant speed of 25 m/s. What is the average speed of the car?
wanted: average speed for the entire trip
given: dtotal = 2.0 km, v1 = 15 m/s, v2 = 25 m/s
relationships:
solution:
Point out that the average velocity is 17 m/s, NOT (15 +25)/2 = 20 m/s. The car spent more time going slower.

45. Assessment
A car travels a total distance of 2.0 kilometers. It travels the first half of the distance at a constant speed of 15 m/s. It travels the second half of the distance at a constant speed of 25 m/s. What is the average speed of the car?
Notice that the average velocity was NOT 20 m/s!
The car spend more time going slower, so the average velocity was less than 20 m/s.