Speed and velocity Interactive calculators are used to build qualitative and quantitative understanding of the speed and velocity equations, including the concepts of positive and negative velocity and the comparison with positive and negative position. The lesson builds a conceptual understanding for the units of m/s, and then presents the speed and velocity equations. Students act out positive and negative positions and velocities to build comprehension.

Objectives Describe one dimensional motion using equations for speed and velocity. Analyze one dimensional motion using equations for speed and velocity. Define and identify positive and negative velocities. The lesson objectives clarify what students should know and be able to do.

Assessment A swallow moves 80 meters in 5.0 seconds. What is its speed? What is its velocity? How can the two be different? Write an English sentence that means the same as this equation: Give an example of an object with positive position and negative velocity. These formative assessments will appear again at the end of the lesson, along with the answers.

Physics terms speed velocity New vocabulary will be defined and reinforced throughout the lesson.

Equations The speed is the distance traveled divided by the time taken. The velocity is the change in position divided by the change in time.

Speed versus velocity Are speed and velocity just two different words for the same thing? In everyday life you probably use the words speed and velocity interchangeably. In physics class, speed and velocity are related, but not exactly the same. Ask students to use the words speed or velocity in a sentence.

Exploring the ideas Click the first interactive calculator on page 78 on speed. This interactive simulation helps students use the equation for speed three different ways.

The speed equation A car travels 30 meters in a trip that lasts 2.0 seconds. What is the car’s speed? 30 2.0 Speed This slide provides an example of solving this equation for speed. To enter data, students need to highlight an entry, click [Clear], enter the new value using the keyboard, then click [Enter].

The speed equation A car travels 30 meters in a trip that lasts 2.0 seconds. What is the car’s speed? 15 m/s 30 15 2.0 Speed Click [Run] to see the car act out the meaning of the equation. Point out that the units in the equation shown work also.

The speed equation If you go a distance of 45 meters at a speed of 16 m/s, how long does this trip last? What variable are you solving for? 45 16 Now students solve the speed equation for time.

The speed equation If you go a distance of 45 meters at a speed of 16 m/s, how long does this trip last? 2.8 seconds 45 16 2.81 Time Point out that the units in the equation shown work also.

The speed equation A fox runs at a speed of 9.7 m/s for 12 seconds. How far does the fox run? 9.7 12 Now students solve the speed equation for distance.

The speed equation A fox runs at a speed of 9.7 m/s for 12 seconds. How far does the fox run? 116 meters 116.4 9.7 12 Distance Point out that the units in the equation shown work also.

How fast is fast? See if you can come up with an example of when an actual object might move at each speed. 0.1 m/s 1 m/s 10 m/s 100 m/s 1000 m/s Because students are more familiar with units of mile per hour, they need to work at developing a feel for values like 1 m/s or 10 m/s so they will know if data or solutions containing speed information is reasonable.

How fast is fast? See if you can come up with an example of when an actual object might move at each speed. 0.1 m/s about 0.22 mph (10 cm/s), the tip of the second hand on your clock 1 m/s 10 m/s 100 m/s 1000 m/s

This is an excellent benchmark to remember! How fast is fast? See if you can come up with an example of when an actual object might move at each speed. 0.1 m/s about 0.22 mph (10 cm/s), the tip of the second hand on your clock 1 m/s 2.2 mph, a slow walk 10 m/s 100 m/s 1000 m/s This is an excellent benchmark to remember!

How fast is fast? See if you can come up with an example of when an actual object might move at each speed. 0.1 m/s about 0.22 mph (10 cm/s), the tip of the second hand on your clock 1 m/s 2.2 mph, a slow walk 10 m/s 22 mph, a brisk bike riding speed 100 m/s 1000 m/s Point out that as a rough estimate, the value in m/s is roughly half of the value in mph.

How fast is fast? See if you can come up with an example of when an actual object might move at each speed. 0.1 m/s about 0.22 mph (10 cm/s), the tip of the second hand on your clock 1 m/s 2.2 mph, a slow walk 10 m/s 22 mph, a brisk bike riding speed 100 m/s 220 mph, a supercar’s top driving speed 1000 m/s

How fast is fast? See if you can come up with an example of when an actual object might move at each speed. 0.1 m/s about 0.22 mph (10 cm/s), the tip of the second hand on your clock 1 m/s 2.2 mph, a slow walk 10 m/s 22 mph, a brisk bike riding speed 100 m/s 220 mph, a supercar’s top driving speed 1000 m/s 2,200 mph, about the F14 fighter jet’s top speed

Speed speed Distance is always positive. Time is always positive. Therefore, speed is always positive! So how do we tell the difference between moving backward and forward?

Speed speed Distance is always positive. Time is always positive. Therefore, speed is always positive! So how do we tell the difference between moving backward and forward? We need a new variable: velocity!

Velocity velocity The velocity is defined as the change in position divided by the change in time. Velocity has direction as well as magnitude.

Velocity velocity The symbol Δ translates to “the change in.” The velocity is defined as the change in position divided by the change in time. The symbol Δ translates to “the change in.” If x = position then Δx means “the change in position.”

Velocity velocity What does Δt mean? The velocity is defined as the change in position divided by the change in time. What does Δt mean?

Velocity velocity What does Δt mean? The velocity is defined as the change in position divided by the change in time. What does Δt mean? If t = time then Δt means “the change in time.”

Velocity velocity The velocity is defined as the change in position divided by the change in time. A change in position, Δx, can be positive or negative. That means that velocity can be positive or negative. The sign of the velocity comes from the sign of the change in x.

Velocity velocity The velocity is defined as the change in position divided by the change in time. A change in position, Δx, can be positive or negative. That means that velocity can be positive or negative. Moving forward is a positive velocity. Moving backward is a negative velocity.

Exploring the ideas Click the interactive calculator on page 78 on velocity.

The velocity equation A car starts at 30 m and finishes at 10 m in a trip that takes 2.0 seconds. Notice: The change in position is -20 m. 10 30 -20 What is the car’s velocity? 2 2 Velocity Enter your initial and final values here. Point out how to enter value.

The velocity equation A car starts at 30 m and finishes at 10 m in a trip that takes 2.0 seconds. Notice: The change in position is -20 m. 10 30 -20 -10 What is the car’s velocity? -10 m/s Click [Run] and see the car drive backwards. 2 2 Velocity The car in the interactive calculator shows negative velocity by driving backwards, to the left. Negative velocity means “headed in the negative direction” regardless of position.

The velocity equation A car travels at 15 m/s. Four seconds after starting out, it is at a position of 40 m. What is the car’s change in position? Where did the car start from? 40 15 4 4 Change in position

The velocity equation A car travels at 15 m/s. Four seconds after starting out, it is at a position of 40 m. What is the car’s change in position? +60 m Where did the car start from? -20 m 40 -20 60 15 4 4 Change in position

The change in position What is the change in position? The variable x stands for position. Subscript “i” means “initial” and subscript “f” means “final”. These next slides elaborate on the definition of velocity.

The change in position What is the change in position? Δx = xf – xi = (8 m) – (2 m) = 6 meters

Velocity What is the velocity if the change in time (Δt) is 2 seconds?

Velocity The velocity is +3 m/s. What is the velocity if the change in time (Δt) is 2 seconds? The velocity is +3 m/s.

What if the person STARTS at 8 m and moves to the left? What is the velocity if the change in time (Δt) is 2 seconds?

What if the person STARTS at 8 m and moves to the left? What is the velocity if the change in time (Δt) is 2 seconds? The man is heading in the negative direction so he has a negative velocity.

Velocity is a vector Velocity can have negative or positive values. The sign of the velocity tells you the direction of motion. Velocity is a vector. A vector is a type of variable which includes directional information in a mathematically useful way.

- + Positive and negative positions and velocities Can there be positive position and negative velocity? I need a volunteer to show me how. - + The next group of slides go carefully through all the possible sign combinations for position and velocity. Students are often confused by the meaning of negative velocity and need to work this through in detail. If possible, draw the origin and the positive and negative directions on the side board so a volunteer can act out the motion for the class.

- + Positive and negative positions and velocities Positive position Can there be positive position and negative velocity? Positive position - +

- + Positive and negative positions and velocities Positive position Can there be positive position and negative velocity? Positive position Negative velocity - + Having students act out these four cases can make a difficult concept clear and enjoyable. Kinesthetic learners will appreciate the chance to put the idea into action.

- + Positive and negative positions and velocities Can there be negative position and positive velocity? I need a volunteer to show me how. - +

- + Positive and negative positions and velocities Negative position Can there be negative position and positive velocity? Negative position - +

- + Positive and negative positions and velocities Negative position Can there be negative position and positive velocity? Negative position Positive velocity - +

- + Positive and negative positions and velocities Can there be negative position and negative velocity? I need a volunteer to show me how. - +

- + Positive and negative positions and velocities Negative position Can there be negative position and negative velocity? Negative position Negative velocity - +

- + Positive and negative positions and velocities Can there be positive position and positive velocity? I need a volunteer to show me how. - +

- + Positive and negative positions and velocities Positive Positive Can there be positive position and positive velocity? Positive position Positive velocity - +

The terminology of motion Tell a story that illustrates how each picture relates to the meaning of each word.

The terminology of motion Tell a story that illustrates how each picture relates to the meaning of each word. For example: A hiker needs to go southwest on a map and this vector represents the hiker’s movement. English language learners benefit from a chance to stop and verbally assimilate and employ new vocabulary.

Assessment A swallow moves 80 meters in 5 seconds. What is its speed? What is its velocity? How can the two be different? The answer to question 1 appears on the next slide. Give students time to formulate an answer.

Assessment A swallow moves 80 meters in 5 seconds. What is its speed? What is its velocity? How can the two be different? Write an English sentence that means the same as this equation: The speed is 16 m/s. The velocity could be either +16 m/s or -16 m/s depending on the bird’s direction. The answer to question 2 appears on the next slide. Ask a volunteer to read their sentence.

Assessment A swallow moves 80 meters in 5 seconds. What is its speed? What is its velocity? How can the two be different? Write an English sentence that means the same as this equation: The speed is 16 m/s. The velocity could be either +16 m/s or -16 m/s depending on the bird’s direction. The velocity is the change in position divided by the change in time.

Assessment Give an example of an object with positive position and negative velocity. This question addresses the third objective: “Define and identify positive and negative velocities.” The answer appears on the next slide.

Assessment - + Negative velocity Positive position Give an example of an object with positive position and negative velocity. A person to the right of the origin but walking to the left would have a positive position and negative velocity. Negative velocity Positive position - +