Vectors and Relative Motion Vector Quantity Fully described by both magnitude (number plus units) AND direction Represented by arrows -velocity -acceleration -force Scalar Quantity Fully described by magnitude (number plus units) alone -mass -temperature

Adding Vectors Vectors in one dimension are added algebraically: 3 m, North+ 4 m, North= 7 m, North 3 m + 4 m = 7 m 3 m, North+ 4 m, South= 1 m, South For a vector-- Sign does not represent value, it represents direction! Traditionally: Up/Right (+) Down/Left (-) 3 m + (-4 m) = -1 m

Adding Vectors in 2 Dimensions- Vectors add Trigonometrically Using Head to Tail Method: 3.0 m 4.0 m 6.5 m 3.0 m 4.0 m 2.2 m 3.0 m m = 6.5 m 3.0 m m = 2.2 m N Vector diagrams show magnitude and direction of vectors and their resultant! 8.0 N N = ?2.0 N ≤ ? ≤ 14 N

Notice Vector Direction: In relation to + x axis

Vector Direction: By agreement, vectors are generally described by how many degrees the vector is rotated from the + x axis 30˚ 150˚ Negative 2D vectors: A - A 180˚ opposite

Resolution (Decomposition) of Vectors If you move a box ˚ from O: 30.0˚ 8.0 m By Geometry: 4.0 m 6.9 m The box has moved– 6.9 m to the right ( +x) These values would be the components of the given vector !

Ø =30.0˚ d = 8.0 m dydy dxdx sinø = dyddyd d y = sinø(d) = sin30.0˚(8.0m) = 4.0 m cosø = dxddxd d x = cosø(d) = cos30.0˚(8.0m) = 6.9 m Adjacent Component! Opposite Component!

V Θ VxVx VyVy Adjacent component: V x = VcosΘ Opposite component: V y = VsinΘ tanΘ = VyVxVyVx Θ = tan -1 (V y /V x ) Be careful of the quandrant! V = √ V x 2 + V y 2

1) A man walks 5.0 km to the East and then walks 3.0 km to the North. What is his displacement from where he started? 2) What are the components of a vector displacement of ˚? 3) If a student walks 56.0 m North and then turns West and walks another 85.0 m, what is his displacement? 4) Vector B has components of d x = -22 m and d y = - 33 m. What is the magnitude and direction of this vector? What is the magnitude and direction of – B ?

5) What is the resultant displacement when a box is moved 5.00 m in the x direction and then m in the y direction? 6) What are the components of the vector shown below? A ø A = 27.3 m Ø = 32.8˚

Adding Vectors Using Components When adding two (or more) vectors, adding the components will give the components of the Resultant vector: A golfer on a flat green putts a ball 7.50 m in the Northeast direction, but misses the hole. He then putts the ball ˚ South of straight East and sinks the putt for a bogey. What single putt would have saved par? d 1 = ˚ d 2 = ˚

d1d1 45.0˚ -38.0˚ d2d2 d 1x = cos45.0˚(7.50 m) = 5.30 m d 1y = sin45.0˚(7.50 m) = 5.30 m d 2x = cos(-38.0˚)(2.30 m) = 1.81 m d 2y = sin(-38.0˚)(2.30 m) = m Head to Tail— On the head of the first goes the tail of the next vector!

d x = d 1x + d 2x = 5.30 m m = 7.11 m d y = d 1y + d 2y = 5.30 m + ( m) = 3.88 m d = √ d x 2 + d y 2 = √(7.11) 2 + (3.88) 2 = 8.10 m Ø = tan -1 (d y / d x ) = tan -1 (3.88 / 7.11) = 28.6˚ d1d1 d2d2 d ø The single (resultant) putt: d = ˚

1) What is the resulting displacement when an object is moved 10.0 m to the North and then 5.0 m to the east? 2) A man leaves his house and walks 6.00 km to the West and then turns and walks 3.50 km to the South. What is his displacement? 3) A woman drives straight East for 65.0 km and then turns 30.0˚ North of East and drives another 33.0 km. What is her displacement? 4) A = ˚ B = ˚ Find the resultant when vector A is added to vector B.

5) Add the following three vectors: A B C ø α β A = 225 m α = 28.0˚ B = 275 m Β = 56.0˚ C = 325 m ø = 15.0˚

Relative Velocity Velocities are vectors and add like vectors: A plane flies through the air at a speed of 255 m/s. The air speed is 33.0 m/s. The velocity of the plane relative to the ground depends upon direction:

In each case, the plane is heading (pointed in that direction) South, but… 288 m/s222 m/s ˚ Remember: Default reference frame is Earth!

A boat travels at 12.0 m/s relative to the water and heads East across a river that flows North at 3.00 m/s. What is the speed and direction of the boat relative to the shore?

V bw = ˚V wg = ˚ V bw V wg V bg ø V bg = (V V 2 2 ) = ( ) = 12.4 m/s Ø = tan -1 (V 2 / V 1 ) = tan -1 (3.00/12.0) = 14.0˚ V bg = ˚

1) A boat heads West across a stream that flows South. What is the velocity of the boat relative to the shore if it heads across with a speed of 8.3 m/s while the water flows South at 2.4 m/s? 3) A barge heading West down a still river travels at 5.0 m/s. A man walks across the barge from North to South at 2.0 m/s. What is the velocity of the man as viewed from a bridge above?

4) A boat wants to travel directly across a river that flows South at 3.0 m/s. If the boat travels at 7.0 m/s in still water, what heading must it take to go straight across? With what speed will the boat travel straight across? 5) An airplane has a velocity of ˚ while flying through a crosswind. What is the heading of the plane? What is the velocity of the wind?

6) A man in a blue car traveling at ˚ views a second red auto traveling at ˚. What is the velocity of the red car relative the the man in the blue car? 7) An airplane flies at ˚ North of east. A second plane flies at ˚ South of North. What is the velocity of the the first plane relative to the second? What is the velocity of the second plane relative to the first?