Vectors. AP/Honors Physics Mr. Velazquez

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1 Vectors AP/Honors Physics Mr. Velazquez

2 The Basics Any quantity that refers to a magnitude and a direction is known as a vector quantity. Velocity, acceleration, force, momentum, displacement Other quantities have no direction associated with them, and can only be represented by a number. These are called scalar quantities. Time, mass, density, temperature, volume, distance** We can use special mathematical arrow notation, v or v, to represent a vector.

3 Adding Vectors If two vectors are in the same direction, adding them is a matter of simple arithmetic. Walk 2 miles north, then 3 miles north, and you ve walked 5 miles north from your starting point If two vectors are acting in opposite directions, one of these vectors will be considered negative, and adding the two is still just a matter of subtraction. Walk 6 miles west, then 2 miles east, and you end up 4 miles west of your starting point This raises the question: what shall we do with vectors that are in two dimensions?

4 Adding Vectors Suppose you walk 3 miles east, then 4 miles north. How far would you then be from your starting point? In terms of vectors, we are adding a displacement vector, D 1 = 3 miles east, to another displacement vector, D 2 = 4 miles north, and asking what the resultant displacement, D R, will be. Graphically, this resultant can be represented by placing one vector onto the endpoint of the other, and drawing a new vector going from the start of the first, to the end of the second. We can find the length of the resultant vector easily in this case, because the vectors are perpendicular, which allows us to use the Pythagorean Theorem D 1 D 2

5 Adding Vectors by Components What if two vectors are not perpendicular to each other? Suppose you walk 6 miles in a direction 30 degrees north-ofeast, then 5 miles at 20 degrees south-of-east. How far (and in what direction) would you be from your starting point? We can solve this graphically, but that would be tedious. Instead, we use a clever method that involves separating each individual vector into its component vectors, and then simply adding up the components. Using a little trigonometry, this process can be simplified and generalized for any number of two-dimensional vectors.

6 Adding Vectors by Components Always remember SOHCAHTOA sin θ = opp hyp = a h cos θ = adj hyp = b h θ tan θ = opp adj = a b

7 Adding Vectors by Components Rearranging these equations: a = h sin θ b = hcos θ θ h = a 2 + b 2 θ = tan 1 a b

8 Adding Vectors by Components V x Applied to a vector, V: V y = V sin θ V y V V y V x = V cos θ V = V x 2 + V y 2 θ V x θ = tan 1 V y V x Alternatively: θ = sin 1 V y V θ = cos 1 V x V

9 Adding Vectors by Components v Suppose the vector in question is a velocity vector of 15.0 m/s applied at an angle of 80 above the horizontal. v y The 15.0 m/s refers to the magnitude of the velocity vector (sometimes referred to as speed ), and is written as v. The 80 is the angle of application, often represented with the greek letter θ. 80 v x According to the trigonometric equations we just derived, we can find the horizontal and vertical components of this vector using the following process: v x = v cos θ = v y = v sin θ = 15 m s cos 80 = m s 15 m s sin 80 = m s

10 Adding Vectors by Components

11 Adding Vectors by Components Given: A = 3.00 m s, B = 6.00 m s, C = 2.50 m s Find the resultant velocity vector, v R First, we break each vector up into its horizontal (x) and vertical (y) components: A x = A cos θ A = A y = A sin θ A = 3.00 m s cos 130 = 1.93 m s 3.00 m s sin 130 = 2.30 m s B x = B cos θ B = B y = B sin θ B = 6.00 m s cos 45 = m s 6.00 m s sin 45 = 4.24 m s C x = 2.50 m s C y = 0

12 Adding Vectors by Components Given: A = 3.00 m s, B = 6.00 m s, C = 2.50 m s Find the resultant velocity vector, v R Now add up all the x-components, then add up the y-components: v Rx = A x + B x + C x v Rx = 1.93 m s m s m s v Rx = m s v Ry = A y + B y + C y v Ry = 2.30 m s m s + 0 m s v Ry = m s We now have the components of v R

13 Adding Vectors by Components Given: A = 3.00 m s, B = 6.00 m s, C = 2.50 m s Find the resultant velocity vector, v R v R Now that we have the components of v R, we can put them together to find the magnitude and angle of the resultant vector v R : v R = v Rx 2 + v Ry 2 = 4.81 m s m s 2 v R = = m s θ = tan 1 v Ry = tan v Rx 4.81 = We now write our resultant as v R = m s, (these are known as polar coordinates )

14 Try it Out! Add the following pairs of vectors, A and B, by adding up their horizontal and vertical components. (Express the resultant vector, R, in polar coordinates) (1) A = 5.00 m s, 30 B = 3.50 m s, 290 (2) A = 27.0 m s, 110 B = 14.5 m s, 15 R 1 = m s, R 2 = m s, 77. 6

15 More to Consider Subtraction is merely the addition of a negative. This is the same for vectors. To subtract a vector, just add its opposite (same magnitude, opposite direction) Take extreme care when determining the angle of application, θ. Occasionally, the angle in the problem might be given in relation to the vertical axis rather than the horizontal. δ θ θ = 90 δ When this happens, just shift your perspective by 90, and/or use SOHCAHTOA to orient yourself.

16 Real World Problem An airplane trip involves three legs, with two stopovers. The first leg is 620 km due east; the second leg is 440 km at 45 south-ofeast; and the third leg is 550 km at 53 south-of-west. What is the plane s total displacement? D 1 = 620 km, D 2 = 440 km, 45 D 3 = 550 km, 233

17 Real World Problem An airplane trip involves three legs, with two stopovers. The first leg is 620 km due east; the second leg is 440 km at 45 south-of-east; and the third leg is 550 km at 53 south-of-west. What is the plane s total displacement? D R D 1 45 D 2 53 D 3 Vector x-component y-component D km 0 km D 2 D 3 D x2 = 440 cos 45 D x2 = 311 km D x3 = 550 cos 233 D x3 = 331 km D y2 = 440 sin 45 D y2 = 311 km D y3 = 550 sin 233 D y3 = 439 km D R 600 km 750 km

18 Real World Problem An airplane trip involves three legs, with two stopovers. The first leg is 620 km due east; the second leg is 440 km at 45 south-of-east; and the third leg is 550 km at 53 south-of-west. What is the plane s total displacement? θ D 1 45 D R = D x 2 + D y 2 = km D R = 960 km D 2 D R 53 D 3 θ = tan 1 D y = tan D x 600 θ = 51 D R = 960 km, 51

19 Classwork/Exit Ticket You ride your bike north 13.0 miles, then you ride in a direction north-of-east for 25.0 miles, then finally you ride in a direction 15 south-of-west for 18.5 miles. What is your final displacement from your starting point (write your answer in polar coordinate form)? D 1 D D 3

Kinematics in Two Dimensions; Vectors

Kinematics in Two Dimensions; Vectors Vectors & Scalars!! Scalars They are specified only by a number and units and have no direction associated with them, such as time, mass, and temperature.!! Vectors