Rec03-Miners Coffin_VectorAddition_2023

Phys 211 1 VECTORS: ESCAPE FROM MINER’S COFFIN Reading: Knight Chapter 3 Objective: By the end of this recitation, you should know the difference between position and displacement, how to add and subtract vectors graphically (in 2D), add and subtract vectors using components (in 3D), and calculate the magnitude of vectors using the Pythagorean theorem. Section #______________ Names: Recorder:_________________________ ___ Checker:__________________________ __ Discussion Leader: ____________________________

Phys 211 2 EXERCISE 1: VECTOR ADDITION • A position vector 𝑟𝑟⃗ 𝐴𝐴 is the vector that gets you (“as the crow flies”) from the origin to a particular point A. A position vector starts at the origin. • A displacement vector Δ𝑟𝑟⃗ 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 is the vector that gets you from point A to point B (it does not generally start at the origin unless point A is the origin); in terms of vectors, it’s the vector difference in the two positions, Δ𝑟𝑟⃗ 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 = 𝑟𝑟⃗ 𝐴𝐴 − 𝑟𝑟⃗ 𝐴𝐴 (note that it’s always the final position minus the initial position). Q1. Vector background questions: 1. On the diagram below (left), draw Δ𝑟𝑟⃗ 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 2. Adding the vector −𝑟𝑟⃗ 𝐴𝐴 is the same as subtracting the vector 𝑟𝑟⃗ 𝐴𝐴 , as shown below. So you can subtract two vectors using vector addition: Δ𝑟𝑟⃗ 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 = 𝑟𝑟⃗ 𝐴𝐴 + ( −𝑟𝑟⃗ 𝐴𝐴 ) . On the right side of the diagram below, add −𝑟𝑟⃗ 𝐴𝐴 to 𝑟𝑟⃗ 𝐴𝐴 (head-to-tail) and use that to draw Δ𝑟𝑟⃗ 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 . EXERCISE 2: TWO-DIMENSIONAL ANALYSIS WITH VECTOR ARROWS Backstory READ THIS: The cave entrance you reach is at surface level and surrounded by level ground 200 meters in every direction. Once inside the cave you use a compass, a plumb bob (i.e. a mass hanging from a string), and a tape measure to determine your displacement vector Δ𝑟𝑟⃗ between landmarks in the cave, recording vector components with respect to the three perpendicular directions: East/West, North/South, and Up/Down. Leg 1. You strike out from the cave Entrance (E) and find yourself in a room full of beautiful stalactites, so you call it Stalactite junction (S) . While you are there, you figure out your displacement from the cave Entrance , and call it Δ𝑟𝑟⃗ 𝐸𝐸𝐴𝐴𝐴𝐴𝐸𝐸 . Leg 2. From Stalactite junction (S) you follow a path to a chamber full of crystal structures: Crystal chamber (C) . You work out that your displacement from the Stalactite junction was Δ𝑟𝑟⃗ 𝐸𝐸𝐴𝐴𝐴𝐴𝑆𝑆 .

Phys 211 4 Q4. Write an equation for your current position ( 𝑟𝑟⃗ 𝐶𝐶 ) in terms of each of the displacement vectors Δ𝑟𝑟⃗ 𝐸𝐸𝐴𝐴𝐴𝐴𝐸𝐸 , Δ𝑟𝑟⃗ 𝐸𝐸𝐴𝐴𝐴𝐴𝑆𝑆 , and Δ𝑟𝑟⃗ 𝑆𝑆𝐴𝐴𝐴𝐴𝐶𝐶 ( write in terms of vector notation, not components !). Q5. You need to dig your way to either the cave Entrance (E) or to the Stalactite junction (S). Thus, you want to figure out which location (E or S) is closer to your current position in Miner’s coffin (M) . A displacement vector Δ𝑟𝑟⃗ 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 is the direct straight-line path that gets you from point A to point B. On your map above (right hand side) , mark three locations (E, S and M; remember C is blocked) and then draw and label the displacement vectors from Miner’s coffin to your two choices of location (e.g., Δ𝑟𝑟⃗ 𝐶𝐶𝐴𝐴𝐴𝐴𝐸𝐸 ). Q6 Another way to find Δ𝑟𝑟⃗ 𝐶𝐶𝐴𝐴𝐴𝐴𝐸𝐸 and Δ𝑟𝑟⃗ 𝐶𝐶𝐴𝐴𝐴𝐴𝐸𝐸 , the two displacement vectors pertinent to your survival, is with vector subtraction of the position vectors 𝑟𝑟⃗ 𝐶𝐶 , 𝑟𝑟⃗ 𝐸𝐸 , and 𝑟𝑟⃗ 𝐸𝐸 . Recall that a displacement vector Δ𝑟𝑟⃗ 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 from point A to point B is also the vector difference in the two positions, Δ𝑟𝑟⃗ 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 = 𝑟𝑟⃗ 𝐴𝐴 − 𝑟𝑟⃗ 𝐴𝐴 , and that adding the vector −𝑟𝑟⃗ 𝐴𝐴 is the same as subtracting the vector 𝑟𝑟⃗ 𝐴𝐴 , as shown below. Draw 𝑟𝑟⃗ 𝐶𝐶 , 𝑟𝑟⃗ 𝐸𝐸 , and 𝑟𝑟⃗ 𝐸𝐸 , and then find Δ𝑟𝑟⃗ 𝐶𝐶𝐴𝐴𝐴𝐴𝐸𝐸 and Δ𝑟𝑟⃗ 𝐶𝐶𝐴𝐴𝐴𝐴𝐸𝐸 below using vector subtraction of the position vectors. (Do they match what you drew in question 5? - explain) 9. Which location should you tunnel towards? If the length of Δ𝑟𝑟⃗ 𝐸𝐸𝐴𝐴𝐴𝐴𝐸𝐸 is about 30 m, estimate how far and in what direction you need to dig (just “eyeball” it – don’t use a calculator or ruler.) CHECK WITH AN INSTRUCTOR AT THIS POINT

Phys 211 5 EXERCISE 3: THREE-DIMENSIONAL ANALYSIS WITH VECTOR COMPONENTS (USE NUMBERS IN THIS PART) We have been working just in two-dimensions, but of course in a cave you would also be moving vertically. Here are three-dimensional displacement vectors for each leg written in component form. We’ll use a coordinate system in which + 𝑥𝑥 is east, + 𝑦𝑦 is north, and + 𝑧𝑧 is. Apply what you learned in 2D to solving this problem. Leg of Trip Displacement (E, N, Up) cave Entrance to Stalactite junction ∆ r E-to-S : (30 m) 𝚤𝚤̂ + (10 m) 𝚥𝚥̂ − (30 m) 𝑘𝑘 � Stalactite junction to Crystal chamber ∆ r S-to-C : ( − 20 m) 𝚤𝚤̂ + (20 m) 𝚥𝚥̂ + (10 m) 𝑘𝑘 � Crystal chamber to Miner's coffin ∆ r C-to-M : ( − 10 m) 𝚤𝚤̂ − (10 m) 𝚥𝚥̂ − (10 m) 𝑘𝑘 � 8. What is your current position (in Miner’s coffin ) in component form? 9. What displacement vector (in component form) would get you from Miners Coffin to Stalactite junction ? 10. Which location ( cave Entrance or Stalactite junction ) is closer for digging a tunnel to? How far do you have to dig? (In three-dimensions, the Pythagorean theorem is 𝑑𝑑 2 = 𝑎𝑎 2 + 𝑏𝑏 2 + 𝑐𝑐 2 .) NOTE : Now that you are in 3D your answer may differ from what you had in 2D (Q7) 11. In approximately what compass direction will you aim your tunnel? (For Quadrant I between due N and due E, the options would be N, N-NE, NE, E-NE, or E; see the Compass Rose to the side, the directions are different for the other quadrants, your answer could be in any quadrant). 12. At approximately what angle “up” above the horizontal (i.e. above the 2D NS/EW plane and into the z-axis) do you need to aim your tunnel? “Compass Rose” FAA (Public domain image)