Unit 1 Continued: Vectors Remember: Scalar the expression of a physical quantity using only magnitude and unit. Ex. 120 km, 56 N, +125 µc, 37 km/h Vector express magnitude, unit and direction. Ex. 21.1 km [N], 56N [R], 37 km/h [E 23 o S] We will be using vectors in this unit to express distances and rates of motion ***You will need a ruler and protractor for this!!
Example 1 Billy jogs 2.0 km [E], then 3.0 km [W] in a total of 25 minutes. a. Sketch a number line. b. Determine the total distance traveled. c. What is Billy s displacement? d. What is Billy s average speed (km/h)? e. What is Billy s average velocity (km/h)?
Answers A. Number line *notice the resultant is a double arrow!! d B. C. total 1 2 d d total total d d 2.0 3.0 5.0km d 5.0 v 12km / t 0.42 d. e. h or d d d d d ( 2 d 1.0) 1 1.0km d 1.0 v t 0.42 or2.4km / h[ W ] (0) 1.0km[ W ] 2.4km / h
Example 2 Betty is dropped off 4.0 km [N] of Corner Brook. She hikes 8.0 km[n], then 12.0 km [S] in 3.0 hrs. a. Sketch a number line for the trip. b. d=? d=? c. v=? v=?
Drawing Vectors (scale diagrams) How could we construct a diagram which communicates the following vectors? (Use a ruler/protractor) A) 13 km [W] B) 25 km/h [S] C) 13 km [W 25 o N] D) 15 km/h [N] + 15 km/h [N] E) 25 km [E] + 15 km [E] F) 25 km [E] + 15 km [E 40 o S] Why do the last two answers differ?
E) F) We are certain the answer in E is 40 km, but what is the resultant in F? To answer that question we will need to learn more about terrifying triangles!!!
Make sure to include an indication of your scale ex. 1cm = 1km Also include the direction system ex.
Drawing Scale Diagrams When we add some 2-D vectors the answer can be drawn using a protractor and a ruler in a scale diagram. Use a scale diagram to find the sum of the following vectors ex1. 30. km [E] + 10. km [N] d=32 km [E20N]
Example 2 What is the resultant velocity of the aircraft? v1 = 180 km/h[w], v2=85km/h [S] Compare your answer with a calculated answer! (pythagoras, soh cah toa)
Example 3 What is the final displacement of two hikers who travel 15 km [E25S] the first day, then 15 km [S40W]? 18km[S11S]
Adding Vectors- An example of Vector Composition One Dimension (1-D) Example 1:A bunny hops 3.0m [E] then 5.0 m [W]. What is its final displacement? We can use a number line.
Practice Adding vectors in Two Dimensions -an example of Vector Composition Example 2: What is the resultant displacement of a hiker that walks 3.0 km [E] then 4.0 km[n]? A. Construct a scale diagram. B. What is the measured resultant displacement? (magnitude and direction)
Example 2 Continued C. How could you calculate this displacement? (magnitude and direction) What? 3 + 4 = 5?!? Wait till I tell Mom! D. Would it matter if the hiker went North first then East? Practice: p. 84 #1 and p. 114 #15
Adding vectors in Two Dimensions -an example of Vector Composition
Ninja Quiz Practice Find the resultant for each. 1. 4.0 km[n] + 6.0km [E] (compare scale diagram to calculated answer) Answer 7.2 km [N56E] 2. 7.0 km [S] + 5.0 km [W] (scale diagram and calculated) Answer 8.6 km [S35W]
Ninja Quiz Practice 1. 3.0 km [S] + 5.0 km [W25 o N] (scale diagram only) 2. 4.0 km [W37 o S] + 5.0 km [N42 o E] 3. 15 m/s [S12 o E] + 29 m/s [N19 o E]
The terrifying triangle... All triangles are TERRIFYING!! It takes a GENIUS to figure out where The angle is measured, where the hippopotamus is located, let alone the mysterious adjacent or opposite sides.
It gets worse... Now we are going to say ANY vector at an angle can actually be made into a triangle!!! (pure insanity) Before you get all upset, let s try a simple example take a deep breath Example 3. A hiker walks 16 km on a heading [E 30 o N]. a. Sketch picture b. How far east did they travel? (x-direction) c. How far north did they travel?(y- direction)
Answers Using soh cah toa In this case, the y-component is on the opposite side of the triangle, we would use sinθ = dy/16km dy = 16sin30 = 8.0 km [N] dx = 16cos30 = 13.9 km [E] ** Do not think you will always find the y-component by using sin
We could arrive at the same location as the hiker if we walked 13.9 km [E] then 8.0 km [N]. Check answer by using pythagorean theorem. Practice
Could we add two vectors at an angle? Easy to draw, tricky to calculate. Ninja Quiz
Can this approach be applied to other Vectors? Forces: Tugboat Example!
Vector Addition Vector Composition is accomplished by adding vectors to find the resultant. We have done two types: 1. Vectors at right angles scale diagram and calculate answer 2. Vectors which are not at right angles scale diagram only
Relative Velocities (1-Dimensional) Describes the velocity of one object compared to another. Interesting Cases: moving sidewalk, river, wind Have you ever been to Pearson International airport in Toronto? Tons of fun on moving sidewalks.
The Relative Velocity of an object depends on your Frame of Reference (your viewpoint) during the event. V og V om V og velocity of object relative to the ground V om velocity of object relative to the medium V mg velocity of medium relative to the ground V mg
A Calgary senior citizen drove his brand new red Corvette convertible out of > the dealership. > > Taking off down the road, he floored it to 130 km/h, enjoying the wind > blowing through what little hair he had left. > > 'Amazing,' he thought as he flew down the Trans-Canada towards Banff, > pushing the pedal even more. > > Looking in his rear view mirror, he saw a Royal Canadian Mounted Police > patrol car behind him, blue and red lights flashing. He floored it to > 160Km/h, then 180, then 200. > > Suddenly he thought, 'What am I doing? I'm too old for this,' and pulled > over to await the RCMP's arrival. > > Pulling in behind him, the Officer walked up to the Corvette, looked at his > watch and said, 'Sir, my shift ends in 30 minutes. Today is Friday. If you > can give me a reason for speeding that I've never heard before, I'll let you > go.' > > The gentleman paused, then said, 'Years ago, my wife ran off with an RCMP > officer. I thought you were bringing her back.' ' > > Have a good day, Sir,' replied the Officer.
One Dimensional Relative Velocity Know 3 cases on p. 97-98 Drawing Vog using vectors Good stories Radar gun, PEI swimming race
Two Dimensional- Relative Velocity Example 1: Tarzan can swim 2.0 m/s in a river which is flowing 1.0 m/s [E]. a. What position must his body be in to swim directly [N]. (sketch pic) [N30 o W] b. What velocity will he cross the river? 1.7 m/s [N]
Two Dimensional- Relative Velocity Example 2: Tarzana can swim 2.0 m/s and heads [N] into the 1.0 m/s [E] current. (sketch pic). What is her resultant velocity? 2.2m/s [N27 o E] p. 103 #1,2a,2b
Time One Dimension Ex.1 Toa Gali (Toa of Water) swims 3.0m/s [N]. The river flows at 1.0m/s [S]. How long does it take her to swim 100.m? What if she swam downstream?
Time Two Dimensions While we will describe the time it takes to cross the river, the same approach can be applied to similar situations ex. Flying in the wind, walking on sidewalk and others.
Ex. 2 Toa Tahu (Toa of Fire) surfs across the lava at 3.0 m/s. The lava moves at 1.0 m/s [E]. If he positions himself so that he ends up straight across from where he started, how long will it take him to cross a 100. m river? What if he just surfed straight [N]?
Finding Time to Cross the River Last Day Tarzan swam across the river (to end up directly across) and Tarzana allowed the current to take her while she headed north. If the river is 200. m across, how long does it take each person to cross? Who wins? Why?
Practice P. 116 #35,36 P.116# 37-41, *43 1-D Practice Practice Sheet Photocopy