Preview Section 1 Introduction to Vectors Section 2 Vector Operations Section 3 Projectile Motion Section 4 Relative Motion

What do you think? How are measurements such as mass and volume different from measurements such as velocity and acceleration? How can you add two velocities that are in different directions? When asking students to express their ideas, you might try one of the following methods. (1) You could ask them to write their answers in their notebook and then discuss them. (2) You could ask them to first write their ideas and then share them with a small group of 3 or 4 students. At that time you can have each group present their consensus idea. This can be facilitated with the use of whiteboards for the groups. The most important aspect of eliciting student’s ideas is the acceptance of all ideas as valid. Do not correct or judge them. You might want to ask questions to help clarify their answers. You do not want to discourage students from thinking about these questions and just waiting for the correct answer from the teacher. Thank them for sharing their ideas. Misconceptions are common and can be dealt with if they are first expressed in writing and orally. Some students will be able to deduce the answer to the first question based on their work with the previous chapter. Some measurements (such as mass and volume) do not include direction, while other measurements (such as velocity and acceleration) do. After discussing this, have students list other types of measurements, and determine whether they each one includes a direction. Students may not be able to answer the second question (unless they have covered this in an earlier science course), but it will help motivate them to learn the upcoming material.

Introduction to Vectors Scalar - a quantity that has magnitude but no direction Examples: volume, mass, temperature, speed Vector - a quantity that has both magnitude and direction Examples: acceleration, velocity, displacement, force Emphasize that direction means north, south, east, west, up, or down. It does not mean increasing or decreasing. Even though the temperature may be going “up”, it is really increasing and has no direction. To further emphasize the distinction, point out that it is meaningless to talk about the direction of temperature at a particular point in time, while measurements such as velocity have direction at each moment.

Vector Properties Vectors are generally drawn as arrows. Length represents the magnitude Arrow shows the direction Resultant - the sum of two or more vectors

Vector or Scalar? Acceleration of a planes take off The number of passengers on the plane The duration of a flight The displacement of the flight The amount of fuel required for the flight

Finding the Resultant Graphically Method Draw each vector in the proper direction. Establish a scale (i.e. 1 cm = 2 m) and draw the vector the appropriate length. Draw the resultant from the tip of the first vector to the tail of the last vector. Measure the resultant. The resultant for the addition of a + b is shown to the left as c. Ask students if a and b have the same magnitude. How can they tell?

Vector Addition Vectors can be moved parallel to themselves without changing the resultant. the red arrow represents the resultant of the two vectors Stress that the order in which they are drawn is not important because the resultant will be the same.

Vector Addition Vectors can be added Vectors can only be added The resultant (d) is the same in each case Vectors can only be added Negative signs indicate…the vector is moving in the opposite direction

Properties of Vectors Click below to watch the Visual Concept.

Sample Resultant Calculation A toy car moves with a velocity of .80 m/s across a moving walkway that travels at 1.5 m/s. Find the resultant speed of the car. Use this to demonstrate the graphical method of adding vectors. Use a ruler to measure the two components and determine the scale. Then determine the size and direction of the resultant using the ruler and protractor. This would make a good practice problem for Section 2, when students learn how to add vectors using the Pythagorean theorem and trigonometry.

Practice Problems A roller coaster moves 85 m, then travels 45 m at an angle of 30.0° above the horizontal. What is its displacement from its starting point ? Use the graphical method. 126 m at 10°

Practice Problems A novice pilots sets a plane’s controls, thinking the plane will fly at 250 km/h to the north. If the wind blows at 75 km/h toward the south-east, what is the plane’s resultant velocity? Use the graphical method. 58.33 m/s at 75 ° north of east

Practice Problems While flying of the Grand Canyon, the pilot slows the plane’s engines down to ½ the velocity in the problem prior. If the wind’s velocity is still 75.km/h toward the south east, what will the plane’s new resultant velocity be? Use the graphical method. 23.61 m/s at 52 ° north of east