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

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

VECTOR BASICS Images: tors/u3l1a.cfm

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

THE RESULTANT IN ONE DIMENSION

The Resultant in Two Dimensions What if our vectors are not in a path of North, East, South, or West? What if our vectors are not in a path of North, East, South, or West? 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 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 Lets Look! Lets Look!

THE RESULTANT IN TWO DIMENSIONS (X AND Y) ors/U3l1b.cfm

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

Calculating Resultants Graphically When determining the resultant graphically you must be careful of several factors: 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. 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 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

GRAPHICALLY DETERMINING A RESULTANT

ANSWERS TO PRACTICE PRACTICE A: km at º W of N OR km at 63.44º N of W PRACTICE A: km at º W of N OR km at 63.44º N of W PRACTICE B: 50 km at 53.13º S of WOR 50 km at 36.87º W of S PRACTICE B: 50 km at 53.13º S of WOR 50 km at 36.87º W of S

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

Example 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. 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 204 km/h at 75° north of east 204 km/h at 75° north of east

Example 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? 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 ZERO ZERO

Distance and Displacement

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

Distance Distance (d) – how far an object travels. Distance (d) – how far an object travels. Does not depend on direction. Does not depend on direction. Imagine an ant crawling along a ruler. Imagine an ant crawling along a ruler. What distance did the ant travel? What distance did the ant travel? d = 3 cm d = 3 cm cm

Distance Distance does not depend on direction. Distance does not depend on direction. Here’s our ant explorer again. Here’s our ant explorer again. Now what distance did the ant travel? Now what distance did the ant travel? d = 3 cm d = 3 cm Does his direction change the answer? Does his direction change the answer? cm

Distance Distance does not depend on direction. Distance does not depend on direction. Let’s follow the ant again. Let’s follow the ant again. What distance did the ant walk this time? What distance did the ant walk this time? d = 7 cm (5cm + 2cm) d = 7 cm (5cm + 2cm) cm

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

Displacement These are the same as the direction used for vectors These are the same as the direction used for vectors Examples of directions: Examples of directions: + and – + and – N, S, E, W N, S, E, W Angles Angles

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

cm Displacement Let’s revisit our ant, and this time we’ll find his displacement. Let’s revisit our ant, and this time we’ll find his displacement. Distance: 3 cm Distance: 3 cm Displacement: +3 cm Displacement: +3 cm The positive gives the ant a direction! The positive gives the ant a direction!

Displacement Find the ant’s displacement again. Find the ant’s displacement again. Remember, displacement has direction! Remember, displacement has direction! Distance: 3 cm Distance: 3 cm Displacement: -3 cm Displacement: -3 cm cm

Displacement Find the distance and displacement of the ant. Find the distance and displacement of the ant. Distance: 7 cm Distance: 7 cm Displacement: +3 cm Displacement: +3 cm cm

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

Speed VS Velocity

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

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

Calculating Speed A car drives 100 meters in 5 seconds. A car drives 100 meters in 5 seconds. What is the car’s average speed? What is the car’s average speed? s = d/t s = d/t s = (100 m) / (5 s) = 20 m/s s = (100 m) / (5 s) = 20 m/s 100 m 1 s2 s3 s4 s5 s

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

Speed A racecar is traveling at 85.0 m/s. How far does the car travel in 30.0 s? A racecar is traveling at 85.0 m/s. How far does the car travel in 30.0 s?

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

Pulling It All Together Back to our ant explorer! Back to our ant explorer! Distance traveled: 7 cm Distance traveled: 7 cm Displacement: +3 cm Displacement: +3 cm Average speed: (7 cm) / (5 s) = 1.4 cm/s Average speed: (7 cm) / (5 s) = 1.4 cm/s Average velocity: (+3 cm) / (5 s) = +0.6 cm/s Average velocity: (+3 cm) / (5 s) = +0.6 cm/s cm s2 s3 s4 s5 s

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

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

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