SCALARS l Scalars only have magnitude l Magnitude

SCALARS l Scalars only have magnitude l Magnitude means length l Example: 50 m

VECTORS Vectors have BOTH magnitude and direction Example: Instead of just 50 m like a scalar, we would say 50 m North or 50 m West

SCALARS AND VECTORS When you combine two or more vectors (with direction) the sum is called the resultant. l Example: Vector 1 is 30 m North Vector 2 is 20 m North The resultant vector is 50 m North

SCALARS AND VECTORS What if one of our vectors is going the opposite direction? Example: Vector 1 is 70 m North Vector 2 is 40 m South The resultant vector is 30 m North

l Let’s look at some vector basics and some more examples

VECTOR BASICS Images: http: //www. physicsclassroom. com/Class/vec tors/u 3 l 1 a. cfm

VECTOR BASICS l When adding two or more vectors together, we always connect the head of Vector 1 to the tail of Vector 2. l When dealing in only 1 -D, it is very similar to adding and subtracting integers

http: //www. physicsclassroom. com/Class/vectors/U 3 l 1 b. cfm THE RESULTANT IN ONE DIMENSION

The Resultant in Two Dimensions l What if our vectors are not in a path of North, East, South, or West? l Once we connect the heads and tails of our vectors, we connect the tail of the first vector to the head of our last vector to find our resultant l Lets Look!

THE RESULTANT IN TWO DIMENSIONS (X AND Y) http: //www. physicsclassroom. com/Class/vecto rs/U 3 l 1 b. cfm

PROPERTIES OF VECTORS l Vectors can be moved parallel to themselves in a diagram l Vectors can be added in any order. For example, A + B is the same as B + A To subtract a vector, add its opposite. SIGNS (DIRECTION) ARE VERY IMPORTANT!!! l For Example: A – B = A + (-B) l

http: //www. physicsclassroom. com/mmedia/vectors/ao. cfm

Calculating Resultants Graphically l When determining the resultant graphically you must be careful of several factors: Scale must be determined and measured accurately with a ruler. Angles (directions) must be done with a protractor. l The resultant is always from the tail of your first vector head of your last vector. Use your ruler and protractor to find the magnitude and direction of the resultant

DETERMINING SCALE http: //www. physicsclassroom. com/Class/vectors/u 3 l 1 a. cfm

GRAPHICALLY DETERMINING A RESULTANT http: //www. physicsclassroom. com/Class/vectors/U 3 l 1 b. cfm

http: //www. physicsclassroom. com/Class/vectors/U 3 l 1 b. cfm

ANSWERS TO PRACTICE A: 11. 18 km at 26. 56 º W of N 11. 18 km at 63. 44º N of W l PRACTICE B: 50 km at 53. 13º S of W 50 km at 36. 87º W of S l OR OR

Example l Which of the following quantities are scalars, and which are vectors? (A) the acceleration of a plane as it takes off (B) the number of passengers on the plane (C) the duration of the flight (D) the displacement of the flight (E) the amount of fuel required for the flight?

Example l A roller coaster moves 85 m horizontally, then travels 45 m at an angle of 30° above the horizontal. What is its displacement from its starting point? (graphical techniques)

ANSWERS • (A) vector (B) scalar (C) scalar (D) vector • (E) scalar 126 m at 10° above the horizontal RESULTANT 30°

Example l A novice pilot sets a plane’s controls, thinking the plane will fly at 250 km/hr to the north. If the wind blows at 75 km/hr toward the southeast, what is the plane’s resultant velocity? Use graphical techniques.

ANSWERS l 204 km/h at 75° north of east

Example l 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?

ANSWER l ZERO

Distance and Displacement

Questions for Consideration l What is distance? l How is displacement different from distance?

Distance l Distance (d) – how far an object travels. l Does l 0 cm l not depend on direction. Imagine an ant crawling along a ruler. 1 2 3 4 5 6 7 8 What distance did the ant travel? ld = 3 cm 9 10

Distance does not depend on direction. l Here’s our ant explorer again. l 0 cm l 1 3 4 5 6 7 8 9 Now what distance did the ant travel? ld l 2 = 3 cm Does his direction change the answer? 10

Distance l l 0 cm l l Distance does not depend on direction. Let’s follow the ant again. 1 2 3 4 5 6 7 8 9 What distance did the ant walk this time? d = 7 cm (5 cm + 2 cm) 10

Displacement l Displacement ( x) – difference between an object’s final position and its starting position. l l Does depend on direction. Displacement = final position – initial position x = xfinal – xinitial ( means change) In order to define displacement, we need directions.

Displacement l In order to define displacement, we need directions. l These are the same as the direction used for vectors l Examples of directions: l l l + and – N, S, E, W Angles

Displacement vs. Distance l Example of distance: l The l Example of displacement: l The l ant walked 3 cm EAST. An object’s distance traveled and its displacement aren’t always the same!

Displacement l Let’s revisit our ant, and this time we’ll find his displacement. - + 0 cm 1 2 3 4 5 6 7 8 Distance: 3 cm l Displacement: +3 cm l l The positive gives the ant a direction! 9 10

Displacement l Find the ant’s displacement again. l Remember, displacement has direction! - + 0 cm 1 2 3 4 5 Distance: 3 cm l Displacement: -3 cm l 6 7 8 9 10

Displacement l Find the distance and displacement of the ant. - + 0 cm 1 2 3 4 5 Distance: 7 cm l Displacement: +3 cm l 6 7 8 9 10

Displacement vs. Distance l An athlete runs around a track that is 100 meters long three times, then stops. l What is the athlete’s distance and displacement? l l l Distance = 300 m Displacement = 0 m Why?

Speed VS Velocity

Speed (s) – Rate at which an object is moving. l We can calculate speed by dividing distance and time l s = d/t l Like distance, speed does not depend on direction. l

Calculating speed l Since speed is a ratio of distance over time, the units for speed are a ratio of distance units over time units.

Calculating Speed l A car drives 100 meters in 5 seconds. 1 s 2 3 4 5 100 m l What is the car’s average speed? ls = d/t l s = (100 m) / (5 s) = 20 m/s

Speed l A rocket is traveling at 10 km/s. How long does it take the rocket to travel 30 km?

Speed l A racecar is traveling at 85. 0 m/s. How far does the car travel in 30. 0 s?

Velocity (v) – speed with direction. l velocity = displacement / time l Remember that displacement is the change in x or x l

Pulling It All Together l Back to our ant explorer! - + 1 s 2 3 4 5 0 cm 1 2 l Distance 3 4 5 6 7 8 9 10 traveled: 7 cm l Displacement: +3 cm l Average speed: (7 cm) / (5 s) = 1. 4 cm/s l Average velocity: (+3 cm) / (5 s) = +0. 6 cm/s

3. 2 Average vs. instantaneous speed Average speed is the total distance traveled divided by the total time taken. l Instantaneous speed is the apparent speed at any moment, such as on a speedometer. l

Instantaneous Velocity l To calculate instantaneous velocity, we want to find the velocity at that specific instant or point in time l Velocity = displacement / time

Average Velocity l To calculate average velocity, we would calculate between two points in time and a given displacement l Velocity = d / t