Vectors v Scalars. Physics 1 st Six Weeks

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1 Vectors v Scalars Physics 1 st Six Weeks

2 An Appetizer to Start...

3 Vectors vs. Scalars In Physics all quantities are in two categories: scalars & vectors. Scalar quantities are described by magnitude (i.e. how much ). Examples of scalar quantities include: speed, mass, volume, distance, and time

4 Vectors vs. Scalars Vector quantities are described by both magnitude and direction. Examples of vector quantities include: force, velocity, acceleration, displacement Vector quantities are often marked with an arrow in equations example: F= m a is Force is equal to Mass x Acceleration

5 Gridiron Physics

6 Vectors A vector is depicted by an arrow with length representing the magnitude and the head of the arrow pointing in the direction of that vector. Vectors are often depicted with scaled vector diagrams. Vector Diagrams show a vector as an arrow drawn to scale in a specific direction

7 Adding Vectors When the directions are the same we simply add the magnitudes together and find the resultant. Vector A in the example to the left has a negative direction relative to Vectors A and B. Therefore when summing the vectors, it would be as if you went forward A and B, then went backwards A. Note: Vectors are added Head to Tail

8 1-D Vector Addition When combining vectors in 1-D, think of graphing in Algebra. To the right and up is considered positive Left and down are considered negative Vectors are always added head to tail

9 Adding 2-D Vectors at Right Angles: The Pythagorean Theorem The Pythagorean Theorem The Pythagorean theorem is a useful method for determining the result of adding two (and only two) vectors that make a right angle to each other. The method is not applicable for adding more than two vectors or for adding vectors that are not at 90-degrees to each other.

10 Examples of finding the Vector Sum with Pythagorean Theorem The sum of vectors is called the Resultant and is depicted as a capital R R 2 = (5) 2 + (10) 2 R 2 = 125 R = (125) R = 11.2 km NW R 2 = (30) 2 + (40) 2 R 2 = 2500 R = (2500) R = 50 km SW

11 The Head to Tail Method : Adding Vectors of Non-right Angles The best method of finding the sum of two vectors that meet at non-right angles is with trigonometry but, if you don t want to do trig (or haven t studied it yet), the next best method is the Head to Tail Method To use the Head to Tail Method, 1. establish a scale (such as 1 cm = 1 mile) 2. draw the vectors to scale, connected head to tail 3. make sure that the angles the vectors point remains unchanged

12 The Head to Tail Method : Adding Vectors of Non-right Angles (continued) 4. Draw the resultant from the tail of the first vector to the head of the last vector. Label this vector as Resultant or simply R. 5. Measure the length of the Resultant & convert it using the scale 6. Measure the angle of the Resultant clockwise from east.

13 Reporting Vector Direction Properly The direction of a vector (when given in degrees) is usually measured counter-clockwise from East Therefore East is used as 0 degrees

14 Head to Tail Method: Example An example of the use of the head-to-tail method is illustrated below. The problem involves the addition of three vectors: 20 m, 45 deg m, 300 deg m, 210 deg. *Note: notice that the angle is measured from the East as 0 degrees counterclockwise

15 Head to Tail: Example continued The head-to-tail method is employed as described above and the resultant is determined (drawn in red). Its magnitude and direction is labeled on the diagram. SCALE: 1 cm = 5 m

16 More on Resultants "To do A + B + C is the same as to do R." *A football player gets hit simultaneously by three defenders (players A, B, and C). *The player experiences three different applied forces. *Each applied force contributes to a total or resulting force. *The three forces are added together using methods of vector addition, & the resultant (vector R) can be determined. *To experience the three forces A, B and C is the same as experiencing force R. *To be hit by players A, B, and C results in the same force as being hit by one player applying force R.

17 Note: it doesn t matter which order the vectors are added in, as long as the length & angle of each is preserved, the Resultant will always be the same.

18 When Adding Vectors the Order Does Not Matter! As long as the magnitude and direction of the vector is maintained they may be added in any order and the resultant will be the same.

19 Distance Vs. Displacement Scientifically, distance is defined as the total amount that an object has traveled. Distance is a scalar quantity However, displacement is the overall change in position from the starting point. This is how far out of place an object is. Displacement is a vector quantity

20 Distance & Displacement Example # 1 A car traveled north 11 km from town A to town B. It then traveled east 11 km to end in town C. Distance traveled was 22 total km Displacement was approximately 15.6 km northeast

21 Distance and Displacement: Example # 2 A runner ran 50 m north on a field. The runner turned and came back south 30 m. Total distance traveled was 80 m. The runner s displacement was 20 m north of the starting point.

22 Distance and Displacement: Example #3 A physics teacher walks 4 meters East, 2 meters South, 4 meters West, and finally 2 meters North. *The physics teacher has walked a total distance of 12 meters, but her displacement is 0 meters. *During the course of her motion, she has "covered 12 meters of ground" (distance = 12 m). *Yet when she is finished walking, she is not "out of place" - i.e., there is no displacement for her motion (displacement = 0).

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