Vector Addition Physics

Vector Addition https://www.youtube.com/watch?v=BLuI118nhzc Start at 0:05 – No Sound

Use it when… Two vectors affect the same object at the same time. Example: Bob is running west at 2.5 m/s, meanwhile the wind is blowing east at 7 m/s.

Vector Addition Steps: Draw the first vector (order doesn’t matter). From the tip, draw the 2nd vector. Answer vector (resultant) connects start to the end.

Examples: You are walking forward moving 3 m/s. You step onto a sliding floor that is moving forward 2 m/s. You continue walking. How fast will you appear to move? Walk 3 m/s Floor 2 m/s S E Answer (Resultant) 5 m/s Forward

Examples: Imagine that you step onto the sliding floor moving the wrong way. How fast will you move now? Walk 3 m/s S Floor 2 m/s Answer (Resultant) 1 m/s Forward E

TWO DIMENSIONS: Imagine that a small bird is flying across the Grand Canyon. Unfortunately, a strong cross wind is blowing him off course. The bird can fly 12 m/s in calm air. The wind is blowing 7 m/s down the canyon. If the canyon is 100 meters wide, then how far off course will the bird be when arrives on the other side? PICTURE VELOCITY TRIANGLE POSITION TRIANGLE TIME FACTOR Wind Bird 12 m/s 100 m S θ 100/12 = 8.33 s Wind 7 m/s Now use v = ∆x ∆t Off course 58.3 m Bird E The time factor is the same for both velocity and position triangles. To solve for the time factor we use similar triangles. ∆x = v∆t ∆x = (7) (8.33) = 58.3 m

TWO DIMENSIONS: How fast is our bird ACTUALLY moving (hypotenuse of velocity triangle)? Use: a2 + b2 = c2 and solve for c: c = √a2 + b2 = √122 + 72 = 13.9 m/s PICTURE VELOCITY TRIANGLE POSITION TRIANGLE TIME FACTOR Wind Bird 12 m/s 100 m S θ 8.33 s Wind 7 m/s 58.3 m Bird E

TWO DIMENSIONS: How far does the bird travel (hypotenuse of position triangle)? Use: a2 + b2 = c2 and solve for c: c = √a2 + b2 = √1002 + 58.32 = 116 m PICTURE VELOCITY TRIANGLE POSITION TRIANGLE TIME FACTOR Wind Bird 12 m/s 100 m S θ 8.33 s Wind 7 m/s 58.3 m Bird E

TWO DIMENSIONS: By what angle is the bird deflected off course? θ= arctan (7/12) = 30.3 degrees PICTURE VELOCITY TRIANGLE POSITION TRIANGLE TIME FACTOR Wind Bird 12 m/s 100 m S θ 8.33 s Wind 7 m/s 58.3 m Bird E

TWO DIMENSIONS: When the bird reaches the other side it faces upwind and flies to its nest. What will be its speed relative to the canyon? How much time will it take to get to the nest? 7 m/s Wind Bird 12 m/s E 5 m/ s S v = ∆x ∆t ∆t = ∆x v = 58.3 m 5 m/s = 11.7 s

TWO DIMENSIONS: When the bird reaches the other side it faces upwind and flies to its nest. What will be the TOTAL time for this story? 8.33 + 11.7 = 20 s

TWO DIMENSIONS: SAME SCENARIO: The bird can fly 12 m/s in calm air. The wind is blowing 7m/s down the canyon. The canyon is 100 meters wide. What direction should our bird face (angle) in order to arrive at the nest directly? In other words, what angel gives a HORIZONTAL resultant? VELOCITY TRIANGLE POSITION TRIANGLE TIME FACTOR Bird 12 m/s Wind 7 m/s θ 100/9.75 = 10.3 s 9.75 m/s S E 100 m b = √c2 - a2 = √122 – 72 = 9.75 m/s

TWO DIMENSIONS: How much time will the bird save by using this strategy? Was the deflection angle (part 1) the same as the correction angle (part 2)? Why? 20 s – 10.3 s = 9.7 s (almost 10 s) θ= arcsin (7/12) = 35.7 degrees (correction) θ= arctan (7/12) = 30.3 degrees (deflection from before) VELOCITY TRIANGLE POSITION TRIANGLE TIME FACTOR Bird 12 m/s Wind 7 m/s θ 100/9.75 = 10.3 s S 9.75 m/s E 100 m Not the same because the hypotenuse has changed

Review: On the test you will be given scenarios and asked to draw vector diagrams to justify your answers. Let’s practice.

The Shark is moving 8 m/s toward me. Examples: I am swimming 2 m/s towards the beach. Meanwhile a shark is chasing me swimming 10 m/s towards the beach. What is the velocity of the shark relative to me? Vector Addition or Vector Subtraction Me 2 m/s Resultant 8 m/s Towards the Beach Shark 10 m/s The Shark is moving 8 m/s toward me.

The jet ski is moving 5 m/s up stream. Examples: A jet ski can move 8 m/s in calm water. It is currently going up a river with a current of 3 m/s. What is the velocity of the jet ski from the POV of someone standing on the shore? Vector Addition or Vector Subtraction River 3 m/s E Jet Ski 8 m/s Resultant 5 m/s up stream S The jet ski is moving 5 m/s up stream.

Examples: A paper plane is thrown 5 m/s to the East.. A fan blows 3 m/s to the South. What’s the velocity of the plane relative to the ground? Vector Addition or Vector Subtraction Plane 5 m/s S θ Fan 3 m/s √52 + 32 = 5.83 m/s E θ = arctan(3/5) = 31 degrees The plane moves 5.83 m/s 31 degrees South of East

Jon is moving 4.47 m/s 26.6 degrees North of West Examples: Jon is rowing his canoe 2 m/s North. Jenny is rowing her canoe 4 m/s East. What is the velocity of Jon from Jenny/s POV? Vector Addition or Vector Subtraction Jon 2 m/s √22 + 42 = 4.47 m/s θ = arctan(2/4) = 26.6 degrees Jenny 4 m/s Jon is moving 4.47 m/s 26.6 degrees North of West