Vectors and Scalars This is longer than one class period. Try to start during trig day.

Number of horses behind the school A SCALAR is ANY quantity in physics that has MAGNITUDE, but NO direction. Scalar Example Magnitude Speed 20 m/s Distance 10 m Age 15 years Heat Number of horses behind the school 1000 calories 0 (now)

Vector Magnitude & Direction Displacement 5 m, NW 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. How are velocity and speed related? Speed is a scalar – it just has magnitude. Velocity has both magnitude and direction. Example: 20 m/s = speed 20 m/s NE = velocity V

Vector Magnitude & Direction Displacement 5 m, NW 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. Any guesses as to what displacement is? (Hint, look at the units!) Displacement is the vector quantity of distance … that is, it tells how far and in what direction two things are located. Vectors are typically illustrated by drawing an ARROW above the symbol. The arrow is used to convey direction and magnitude.

How to draw vectors The length of the vector, drawn to scale, indicates the magnitude of the vector quantity. the direction of a vector is the counterclockwise angle of rotation which that vector makes with due East or x-axis.

How to draw vectors – lady bug displacement displacement x = 6 cm, 250 NOTE 1: Displacement shows how far apart something is now compared to from where it started. It does not show show the path or the total distance travelled. displacement x = 6 cm, 250 length = magnitude 6 cm 250 above x-axis = direction NOTE 2: When drawing a vector you MUST MUST MUST put a ‘head’ or arrow on the vector to demonstrate which direction it is pointing. x L Head 250 H Tail

Quick Review What is the difference between a scalar and a vector? What are the parts of a vector?

Quick Review What is the difference between a scalar and a vector? Scalars have magnitude only, vectors have magnitude and direction. What are the parts of a vector? tail head

Example of a vector velocity of a plane A resultant (the real one) velocity is sometimes the result of combining two or more velocities.

Adding Vectors – Plane example 1 (tailwind) A small plane is heading south at speed of 200 km/h (This is what the plane is doing relative to the air around it) To understand how far the plane is traveling relative to the ground we need to add the two vectors – the plane’s heading and the tailwind. We add vectors by moving them head to tail and finding the resultant (sum). 1. The plane encounters a tailwind of 80 km/h. 80 200 km h 280 km h 200 km h e The resulting velocity relative to the ground is 280 km/h S 80 km/ h

Adding Vectors – Plane example 2 (headwind) A small plane is heading south at speed of 200 km/h (This is what the plane is doing relative to the air around it) 2. It’s Texas: the wind changes direction suddenly 1800. Now the plane encounters a 80 km/h headwind How do we figure out the plane’s velocity relative to the ground? Move the vectors head-to-tail and find the resultant vector. The resultant vector always goes from the tail of the first vector to the head of the second vector 200 km h 80 120 km h e The resultant velocity is only 120 km/h south..

Adding or subtracting vectors in a straight line is easy, but what if the wind is coming from the side? We need to use trigonometry.

3. The plane encounters a crosswind of 80 km/h. Will the crosswind speed up the plane, slow it down, or have no effect? How can we find out? 80 km h 200 As in the other two examples, we have to add two velocity vectors head-to-tail in order to find the resultant vector. How can we calculate the magnitude of the resultant vector? Use trigonometry! Specifically, Pythagorean theorem. 200 km h 80 RESULTANT RESULTANT VECTOR (RESULTANT VELOCITY) v = 215 km/h SE How can I find the exact angle? Use trig! Specifically, tan-1. So the plane is traveling 215 km/h at 22o E of S. F = tan-1 (80 / 200) = 22o

Because direction matters! 200 km h 280 km h 120 km h 215 km h 180 km h So why do we use vectors in physics??? Because direction matters!

The order in which two or more vectors are added does not effect result. vectors can be moved around as long as their length (magnitude) and direction are not changed. Vectors that have the same magnitude and the same direction are the same. Adding A + B + C + D + E yields the same result as adding C + B + A + D + E or D + E + A + B + C.

Are all these vectors equal or not? How do you know? Quick check: Are all these vectors equal or not? How do you know? Yes! They are all equal. A vector quantity is determined by its length and direction. Its position doesn’t matter. That’s why we can move vectors around to add them.

Example: A man walks 54. 5 meters east, then 30 meters west 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 Example: A man walks 54.5 meters east, then again 30 meters east. Calculate his displacement relative to where he started. 54.5 m, E 30 m, E 84.5 m, E Example: A man walks 54.5 meters east, then 30 meters north. Calculate his displacement relative to where he started. 62.2 m, NE 30 m, N 54.5 m, E

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. 62.2 m, NE 30 m, N  = 290 q 54.5 m, E So the COMPLETE final answer is : 62.2 m, 290 or 62.2 m @ 290

Try the following on your own A person walks 5m N then walks 8m S. Calculate their displacement. A ball is thrown 25 m/s E. A tailwind of 5 m/s E is blowing. Calculate the resulting velocity. A boat moves with a velocity of 15 m/s, N in a river which flows with a velocity of 8.0 m/s, west. Calculate the boat's resultant velocity with respect to due north

Try the following on your own. A person walks 5m N then walks 8m S. Calculate their displacement. A ball is thrown 25 m/s E. A tailwind of 5 m/s E is blowing. Calculate the resulting velocity. 5 m 8 m 3m South 3 m 30 m 30m East 25 m 5 m

Calculate the boat's resultant velocity with respect to due north. A boat moves with a velocity of 15 m/s, N in a river which flows with a velocity of 8.0 m/s, west. Calculate the boat's resultant velocity with respect to due north. 8.0 m/s, W 15 m/s, N q The Final Answer :

Example 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. 12 m, W 6 m, S 20 m, N q R 35 m, E 14 m 23 m 𝑅 =27 𝑚 @ 31 0 =27 𝑚, 31 0 The Final Answer:

IMPORTANT NOTE: The Pythagorean theorem and trig functions can only be used for right angle triangles! Later, I will teach you how to handle vectors that meet at angles other than right angles.

Multiplying vector by a scalar Multiplying a vector by a scalar will ONLY CHANGE its magnitude – not direction. A ½ A Opposite vectors A One exception: Multiplying a vector by “-1” does not change the magnitude, but it does reverse it's direction – 3A – A - A

Vector Components Any vector can be “resolved” into two component vectors. How do we calculate Ax and Ay ? Use trig! cos q = 𝐴𝑥 𝐴 sin q = 𝐴𝑦 𝐴 Ax = A cos q Ay = Asin q A Ay q Ax Ax is the horizontal component – or x component -- of the vector. Ay is the vertical component – or the y component – of the vector.

vy vx v = 34 m/s @ 48° . Find vx and vy Example: A plane heads east, while the wind moves a plane north. As a result, the plane moves with velocity of 34 m/s @ 48°relative to the ground. Calculate the plane's heading and wind velocity. What does this mean?? It means we need to find the x-component of the plane’s resulting velocity (= wind velocity) and the y-component of the plane’s resulting velocity (= plane’s heading). v = 34 m/s @ 48° . Find vx and vy cos 48o = 𝑉𝑥 34 𝑚/𝑠 vx = 34 m/s cos 48° = 23 m/s sin 48o = 𝑉𝑦 34 𝑚/𝑠 vy = 34 m/s sin 48° = 25 m/s vy q vx

A plane moves with a velocity of 63.5 m/s at 32 degrees South of East. Calculate the plane's horizontal and vertical velocity components. cos (32 0 ) = 𝑣 𝑥 63.5 𝑚/𝑠 vx = ? 𝑣 𝑥 =63.5 cos (32 0 ) =53.9 𝑚 𝑠 𝐸 320 Vy = ? 63.5 m/s sin (32 0 ) = 𝑣 𝑦 63.5 𝑚/𝑠 𝑣 𝑦 =63.5 sin ( 32 0 ) =33.6 𝑚 𝑠 𝑆

Problems for you to try individually A person walks 450 m @ 120 degrees. Find the x and y component vectors. A car accelerates 6 m/s2 at 40 degrees. Find the x and y component vectors.

Problems for you to try individually A person walks 450 m @ 120 degrees. Find the x and y component vectors. -225 m = Ax 390 m north = Ay A car accelerates 6 m/s2 at 40 degrees. Find the x and y component vectors. 5 m = Ax 4 m = Ay

You can find a vector from its components. This problem may be written differently, but its exactly the same type of problem we did during our first lesson on vectors! Just add the components to find the overall vector! Let: Fx = 4 N Fy = 3 N . Find magnitude and direction of the vector. Fx2+ Fy2 = F2 Fy q Fx  = arc tan (¾) = 370