Vectors and Scalars Physics 1 - L

Scalar Refresher: A SCALAR is ANY quantity in physics that has MAGNITUDE, but NOT a direction associated with it. Magnitude – A numerical value with units. Scalar Example Magnitude Speed 20 m/s Distance 10 m Age 15 years Heat 1000 calories

Vectors cont. Vector Magnitude & Direction Velocity 20 m/s, N Acceleration 10 m/s/s, E Force 5 N, West A VECTOR is ANY quantity in physics that has BOTH MAGNITUDE and DIRECTION. Vectors are typically illustrated by drawing an ARROW above the symbol. The arrow is used to convey direction and magnitude.

Using Vectors It is a good habit to label your vectors in a diagram Always include an arrowhead on your diagram F=10N

Adding Vectors Vectors can be added Using Scale Diagrams- Graphically OR Using Trigonometry (SOH-CAH-TOA) We will start with Graphical Vector Addition

Parallel Vectors VECTOR ADDITION – If 2 similar vectors point in the SAME direction, add them. Example: A man walks 54.5 meters east, then another 30 meters east. Calculate his displacement relative to where he started? + 54.5 m, E 30 m, E Notice that the SIZE of the arrow conveys MAGNITUDE and the way it was drawn conveys DIRECTION. 84.5 m, E

Parallel Vectors cont. VECTOR ADDITION cont. - If 2 vectors are going in opposite directions, you SUBTRACT. Example: A man walks 54.5 meters east, then 30 meters west. Calculate his displacement relative to where he started? 54.5 m, E - 30 m, W 24.5 m, E

Graphical Vector Addition   Adding two vectors A and B GRAPHICALLY, also called, TIP to TAIL METHOD will give the sum of the vectors R ( Resultant Vector ) R is equal to the distance from the beginning to the end point. Arrows are used to represent the vectors vectors will be drawn to scale (EX: 1 cm = 2 m) the beginning of vector B is placed at the end of vector A The vector sum R can be drawn as the vector from the beginning to the end point. ( indicated by the dotted line ) This method can be used with multiple vectors using the tip to tail method to solve for the R Vector. You will use this method to complete the post vector scavenger hunt graphical vector addition activity.

Combining Vectors The individual vectors are called components The “overall” vector is called the resultant.

Graphical Vector addition Use your graph paper wisely! Think about the vectors given and place them in the coordinate system accordingly.

1. Use Tip to Tail method & graph vectors 2 1. Use Tip to Tail method & graph vectors 2. Solve for the Resultant (displacement) 3. Solve for total distance traveled. Graph the vectors using the tip to tail method

Lets learn how to read angles! Pass out Angle worksheet

1st – Review Geometry Quadrants Where is Zero?

Did you correctly id zero Did you correctly id zero? Locate N, S, E, and W Note: 12 u – is a scale for magnitude. 90 Vector A 12 u, 225° OR 12 u, 45° W of S 12 u, 45° S of W 180 ZERO A 12 u 270

Review: Right Triangle Trigonometry SOH CAH TOA sin  = opposite hypotenuse cos  = adjacent hypotenuse tan  = opposite adjacent hypotenuse opposite C B  A adjacent And don’t forget: Pythagorean Theorem A2 + B2 = C2

Perpendicular Vectors When 2 vectors are perpendicular, you must use the Pythagorean Theorem. A man walks 95 km, East then 55 km, north. Calculate his RESULTANT DISPLACEMENT. 55 km, N 95 km,E

Calculator MODE Check Enter mode: Make sure your calculator is in Degrees NOT RADIANS!

BUT…..what about the VALUE of the angle??? Just putting North of East on the answer is NOT specific enough for the direction. We MUST find the VALUE of the angle. Remember: SOH CAH TOA 109.8 km 55 km, N To find the value of the angle we use a Trig function called TANGENT. ( TOA ) q N of E 95 km,E q - Greek symbol Theta FYI: Tan -1 refers to the inverse – Use 2nd TAN on the calculator.

What if you are missing a component? Suppose a person walked 65 m, 25 degrees East of North. What were his horizontal and vertical components? The goal: ALWAYS MAKE A RIGHT TRIANGLE! To solve for components, we often use the trig functions sine and cosine. ( SOH CAH ) 65 m Y = ? 25

Example 23 m, E - = 12 m, W - = 14 m, N 6 m, S 20 m, N 35 m, E R A bear, searching for food wanders 35 meters east then 20 meters north. Frustrated, he wanders another 12 meters west then 6 meters south. Calculate the bear's displacement. 23 m, E - = 12 m, W - = 14 m, N 6 m, S 20 m, N 35 m, E R 14 m, N q 23 m, E The Final Answer:

Vector Addition-Analytical To add vectors mathematically follow the steps: Write - SOH CAH TOA on the top of your paper. Make sure your calculator is in DEGREE MODE. Resolve the vectors to be added into their x- and y- components by making a sketch. Draw the X component – What quadrant is it in? I, II, III, IV What sign will it have? + or - Draw the Y component - What quadrant is it in? I, II, III, IV What sign will it have? + or - Ax – on the x-axis – solve using cosine ( CAH ) Ay – on the y axis – solve using sin ( SOH ) After solving for Ax and Ay – check yourself using the Pythagorean Theorem.

Vector Add. – Analytical cont. Add the x- components together to get the x-component of the resultant (Rx ) by Adding Like Terms : Rx = Ax + Bx Add the y- components together to get the y-component of the resultant (Ry ) by Adding Like Terms: Ry = Ay + By Use the Pythagorean Theorem and the x and y components together to find the magnitude of resultant. 12. Use inverse tangent ( 2nd TAN )to find the angle and then adjust to find the direction of the resultant from Zero East.

Vector Resolution Example Given: vector A at angle  from horizontal. Resolve A into its components. (Ax and Ay) y Hypo A Ay Adj  x Ax Opp Use trig functions: SOH CAH cos =Ax/A so… Ax = A cos sin= Ay/A so… Ay = A sin

Vector Addition-Analytical Given: Vector A is 90 u, 30O and vector B is 50 u, 125O. Find the resultant R = A + B mathematically. Example: B Ax= 90 cos 30O = 77.9 u Ay= 90 sin 30O = 45u Bx = 50 cos 55O =- 28.7u By = 50 sin55O = 41u Note: Bx will be negative because it is acting along the neg x axis. A By Ay 55O 30O Bx Ax First, calculate the x and y components of each vector.

Vector Addition-Analytical cont. Find Rx and Ry: Rx = Ax + Bx Ry = Ay + By Rx = 77.9 - 28.7 = 49.2 u Ry = 41 + 45 = 86 u R Ry  Then find R: R2 = Rx2 + Ry2 R2 = (49.2)2+(86)2 , so… R = 99.1 u To find direction(angle) of R:  = tan-1 ( Ry / Rx )  = tan-1 ( 86 / 49.2 ) = 60.2O Rx

Stating the final answer All vectors must be stated with a magnitude and direction. Angles must be adjusted by adding compass directions( i.e. N of E) or adjusted to be measured from East (or 0° or the positive x-axis). When added graphically, the length of resultant must be multiplied by scale to find the magnitude.