Vectors and Scalars AP Physics C

Scalar A SCALAR is ANY quantity in physics that has MAGNITUDE, but NOT a direction associated with it. Magnitude – A numerical value with units. Scalar Example Magnitude Speed 20 m/s Distance 10 m Age 15 years Heat 1000 calories

Vector Vector Magnitude & Direction 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. .

Polar Notation Polar notation defines a vector by designating the vector’s magnitude |A| and angle θ relative to the +x axis. Using that notation the vector is written: F=12<210 How do we write this vector?

Polar Notation In this picture we have a force vector of 12 Newtons oriented along the -x axis. However, polar notation is relative to the + x axis. Therefore, it would be characterized by Answer: F= 12<180 In this last picture we have 2 vectors. They are characterized by: Answer: C=2<30 D=4<-50 or 4<310 F=12<180 and C=2<30 and D=4<-50 or 4<310

Scalar Multiplication Multiplying a vector by a scalar will ONLY CHANGE its magnitude. How can we change the direction of a scalar? Thus if A = 12 < 105, then what does 2A equal ? Answer: 2A=24<105 Thus if A = 12 < 105, then what does – A equal? Answer: -A= 12<285 OR -A = 12<-75 -1/2A

Unit Vector Notation An effective and popular system used in engineering is called unit vector notation. It is used to denote vectors with an x-y Cartesian coordinate system.

Unit Vector Notation =3j = 4i J = vector of magnitude “1” in the “y” direction = 4i i = vector of magnitude “1” in the “x” direction The hypotenuse in Physics is called the RESULTANT or VECTOR SUM. The LEGS of the triangle are called the COMPONENTS 3j Vertical Component NOTE: When drawing a right triangle that conveys some type of motion, you MUST draw your components HEAD TO TOE. 4i Horizontal Component

Unit Vector Notation J= 2i + 4j K=2i -5j How would you write vectors J and K in unit vector notation? J= 2i + 4j K=2i -5j J=2i + 4j K=2i – 5j

Applications of Vectors VECTOR ADDITION – If 2 similar vectors point in the SAME direction, add them. Example: A man walks 54.5 meters east, then another 30 meters east. Calculate his displacement relative to where he started? ANSWER: 84.5 m + Notice that the SIZE of the arrow conveys MAGNITUDE and the way it was drawn conveys DIRECTION. 84.5 m

Applications of Vectors VECTOR SUBTRACTION - If 2 vectors are going in opposite directions, you SUBTRACT. Example: A man walks 54.5 meters east, then 30 meters west. Calculate his displacement relative to where he started? ANSWER: 24.5 m EAST - 24.5 m east \

Magnitude of a vector The symbol |A| means the magnitude of vector A. The magnitude of a vector is equal to the square root of the sum of the squares of the components of the vector. |A|= √(Ax2 + Ay2 )

Non-Collinear Vectors When 2 vectors are perpendicular, you must use the Pythagorean theorem. |R|=√((95km)2 + (55km)2 R= 109.8 km A man walks 95 km, East then 55 km, north. Calculate his RESULTANT DISPLACEMENT. 55 km, N 109.8 km 95 km,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. Find the value of the angle. Write the resultant vector in the following: Polar Notation Cartesian unit vector form 109.8 km 55 km, N q N of E 95 km,E R = 109.8 km<30 or 109.8 km, 30 degrees North of East Or 95i + 55j 109.8 km<30 or 109.8 km, 30 degrees N of E 95i +55j

What if you are missing a component? Suppose a person walked 65 m, 25 degrees East of North. What were his horizontal and vertical components? H.C. = ? VC = 65cos25 VC= 58.91 j HC=65sin25 HC 27.47i V.C = ? 25 65 m 58.91 j 27.47i

Adding Vectors Algebraically In order to add vectors algebraically, you must do the following: Resolve each vector into components (For vector A, find Ax and Ay. For vector B, find Bx and By. And so on) Add like components to find components of the resultant. (Rx=Ax +Bx +…, Ry=Ay+By+…) Find magnitude of resultant and direction of resultant (|R|= √(Rx2 + Ry2 )

Example 1 12 m, W ANSWER:R =26.93 m, Ө=31.3 degrees Or R=23i + 14j 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 ANSWER:R =26.93 m, Ө=31.3 degrees Or R=23i + 14j 6 m, S 20 m, N 35 m, E R= 26.93 m, θ=31.3 23i + 14j

Example 2 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. Answer: R =17 m/s, Ө=28.1 degrees west of north OR R=-8i + 15j 8.0 m/s, W 15 m/s, N Rv q R = 17 m/s, θ=28.1 degrees west or noth -8i + 15 j

Example 3 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. HC=53.85i VC= -33.64 j H.C. =? 32 V.C. = ? 63.5 m/s 53.85 I – 33.64j

Example 4 Find the sum of two vectors A and B lying in the xy plane. Find A - B A = 2.00i + 3.00j B =5.00i -4.00j ANSWER: A-B = -3i + j

Example 5 A particle undergoes three consecutive displacements: Δr1 = (1.50i + 3.00j -1.20k)cm,Δr2=(2.30i – 1.40j -3.60k)cm, and Δr3=(-1.30i + 1.50j)cm. Find the components of the resultant displacement and its magnitude. ANSWER: R= 2.5i+3.1j -4.8 k |R|=√(2.5)2 +(3.1)2 +(1.5)2 |R|=6.24 cm 2.5i +3.1j-4.8k 6.24 cm

Example 6 A hiker beings a two day trip by walking 25.0 km due southeast from her car. She stops a sets up her ten for the night. On the second day she walks 40.0 km in a direction 60 degrees north of east, at which point she discovers a ranger’s tower. Determine the components of the hikers displacement in each day. Determine the components of the hiker’s total displacement for the trip. a) 17.7i -17.7j and 20.0i +34.6 j b)37.7i +16.9j a) 17.7i-17.7j b) 37.7i + 16.9 j