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Vectors Vector: a quantity that has both magnitude (size) and direction Examples: displacement, velocity, acceleration Scalar: a quantity that has no direction associated with it, only a magnitude Examples: distance, speed, time, mass

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**Vectors are represented by arrows. **

The length of the arrow represents the magnitude (size) of the vector. And, the arrow points in the appropriate direction. 20 m/s 50 m/s NW East

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**Adding vectors graphically**

Without changing the length or the direction of any vector, slide the tail of the second vector to the tip of the first vector. 2. Draw another vector, called the RESULTANT, which begins at the tail of the first vector and points to the tip of the last vector.

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**Adding co-linear vectors (along the same line)**

B = 4 m A + B = R = 12 m C = 10 m/s D = - 3 m/s C + D = 10 + (-3) = R = 7 m/s

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**Adding perpendicular vectors**

11.67 m 6 m 10 m How could you find out the length of the RESULTANT? Since the vectors form a right triangle, use the PYTHAGOREAN THEOREM A2 + B2 = C2

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Vector COMPONENTS Each vector can be described to terms of its x and y components. Y (vertical) component X (horizontal) component If you know the lengths of the x and y components, you can calculate the length of the vector using the Pythagorean.

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**Sometimes, we designate a particular vector by its components:**

A = 3x + 4y A is a vector whose x-component is 3 and y-component is 4. A 4 3

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**To add 2 vectors using components is easy:**

If A = 3x + 4y and B = 2x + 8y Then A + B = 5x + 12y

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**Draw a line from the arrow tip to the x-axis. **

Drawing the x and y components of a vector is called “resolving a vector into its components” Make a coordinate system and slide the tail of the vector to the origin. Draw a line from the arrow tip to the x-axis. The components may be negative or positive or zero. X component Y component

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Calculating the components How to find the length of the components if you know the magnitude and direction of the vector. Sin q = opp / hyp Cos q = adj / hyp Tan q = opp / adj SOHCAHTOA = 12 m/s A Ay = A sin q = 12 sin 35 = 6.88 m/s q = 35 degrees Ax = A cos q = 12 cos 35 = 9.83 m/s

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**Adding Vectors by components**

A = 18, q = 20 degrees B = 15, b = 40 degrees Adding Vectors by components B A a b Slide each vector to the origin. Resolve each vector into its x and y components The sum of all x components is the x component of the RESULTANT. The sum of all y components is the y component of the RESULTANT. Using the components, draw the RESULTANT. Use Pythagorean to find the magnitude of the RESULTANT. Use inverse tan to determine the angle with the x-axis. q A B R x y 18 cos 20 18 sin 20 -15 cos 40 15 sin 40 5.42 15.8 a = tan-1(15.8 / 5.42) = 71.1 degrees above the positive x-axis

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Unit Vectors A unit vector is a vector that has a magnitude of exactly 1 unit. Depending on the application, the unit might be meters, or meters per second or Newtons or… The unit vectors are in the positive x,y, and z axes and are labeled

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Unit Vectors

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**Vector Multiplication**

Scalar product Vector x scalar = vector Example: F = ma “dot” product vector • vector = scalar Example: W = Fd “cross” product vector x vector = vector

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**Magnitude of Cross products: A x B = ABsinq **

Dot products: A • B = ABcosq Magnitude of Cross products: A x B = ABsinq Use the “right-hand rule” to determine the direction of the resultant vector. Multiplication using unit vector notation….

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**Direction of cross products**

i x j = k j x k = i k x i = j j x i = -k k x j = -i i x k = -j + ijkijk -

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**You can only DOT vectors that have colinear components!**

You can only CROSS vectors that have perpendicular components!

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“Relative Motion”

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**You’re driving with a constant velocity of 30 m/s North**

You’re driving with a constant velocity of 30 m/s North. Another car is driving with a constant velocity of 35 m/s North. What is his velocity relative to you? What is your velocity relative to him?

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**Inertial reference frames: A NON ACCELERATING reference frame.**

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To get a position or velocity relative to a particular inertial reference point, you must SUBTRACT away the position or velocity of the reference point (relative to an origin) Examples for two objects, A and B: A: r = 6i – 2j B: r = -3i + 8j What is A relative to B? B relative to A?

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Suppose A: r = 3ti – 4tj B: r = 2i – 6tj Are both of these objects inertial reference frames? How would you know? What is the velocity of A relative to B? What is the velocity of B relative to A?

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**What happens when a boat tries to cross a river**

What happens when a boat tries to cross a river? What is the boat’s velocity relative to the water? What is the boat’s velocity relative to the shoreline? River and Boat

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A boat capable of going 5 m/s in still water is crossing a river 100 m wide with a current of 3 m/s. If the boat points straight across the river, where will it end up- straight across the river or downstream? Downstream, because that is where the current will carry it even as it goes across! How long will it take the boat to cross the river? How far downstream will the boat be? What is the resultant velocity?

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**Sketch the 2 velocity vectors: **

the boat’s velocity, 5 m/s and the river’s velocity, 3 m/s Make two columns, “across” and “downstream”. Write what you know in each direction (d, v, t) Across Downstream d = 100 m d = ? v = 5 m/s v = 3 m/s t = ? t = ? We will use the equation d = vt Time to cross = 100 / 5 = 20 s Distance downstream = 3 m/s x 20 s = 60 m Resultant velocity, 3 m/s 5 m/s Resultant velocity

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**Correcting the Course = 10.64° S of E groundspeed q Windspeed = 12m/s**

An airplane capable of flying at 65 m/s needs to fly 200 km due east but there is a wind blowing from south to north at 12 m/s. At what angle must the airplane be aimed in order to arrive at the desired destination? How long will the trip take? 1. Draw the desired RESULTANT- what you want the airplane to do relative to the ground. (If you’re drawing velocity vectors, this will be the “groundspeed”) 2. Draw the “windspeed” vector along the direction of the wind and pointing to the tip of the groundspeed vector. Draw a vector from your present location pointing to the tail of the windspeed vector that represents where the pilot must steer the airplane. This vector is the “airspeed” of the plane, the speed of the plane relative to the air. The vector addition of : where you aim the plane (airspeed) + what the wind is doing (windspeed) will result in where the plane goes relative to the ground (groundspeed) 4. Use trig functions and/or Pythagorean to find the angle and resulting groundspeed. NOTICE: The RESULTANT of this vector addition is NOT a hypotenuse. 5. The time is found by dividing the distance by the resulting groundspeed. = 10.64° S of E groundspeed q Windspeed = 12m/s Airspeed = 65 m/s

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