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

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

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 The resultant (d) is the same in each case Subtraction is simply the addition of the opposite vector.

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.

Graphical Practice Problems Which of the following quantities are scalars and which are vectors? Acceleration of a plane as it takes off The number of passengers on the plane The duration of the flight Displacement of the flight Amount of fuel required for the flight

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

Graphical Practice Problems A novice pilot sets a plane’s controls, thinking the plane will fly at 250km/h to the north. If the wind blows at 75 km/h toward the southeast, what is the plane’s resultant velocity?

Graphical Practice Problems While flying over the Grand Canyon, the pilot slows the plane’s engines down to one-half the velocity of item 3. If the wind’s velocity is still 75 km/h toward the southeast, what will the plane’s new resultant velocity be?

Graphical Practice Problems The water used in many fountains is recycled. For instance, a single water particle in a fountain travels through an 85 m system and then returns to the same point. What is the displacement of a water particle during one cycle?