1.2.1 Vectors. 1 P age. Examples What is the reference vector angle for a vector that points 50 degrees east of south?

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1 1.2.1 Vectrs Definitins Vectrs are represented n paper by arrws directin = magnitude = Examples f vectrs: Examples What is the reference vectr angle fr a vectr that pints 50 degrees east f suth? What is the reference vectr angle fr a vectr that pints 20 degrees nrth f east? Cmpass Pint System uses angles measured frm a given cmpass directin. Uses tw cmpass directins fr a reference. ngular Systems Reference Vectr System uses due east as a 0 reference. ngles are measured frm that pint. CCW = + CW = - 1) Measure angle θ (a) Practice (b) θ θ (c) θ 20 meters at 10 suth f west 34 meters at 42 east f nrth 20 meters at 34 meters at 1 P age

2 2) dd a vectr t each diagram such that the angle given is between the given vectr and yur drawing. 3) Fr each f the sets f vectrs belw ( and ), sketch what yu think the resultant sum f the tw vectrs will lk like. (Hint: imagine each arrw as a persn pulling with a frce prprtinal t the arrw length!) (a) 45 (a) (b) (b) 125 (c) (c) 300 (d)

3 4) elw are grups f resultant vectrs (R) paired with a single cmpnent (). Sketch the missing vectr that when added t vectr wuld prduce the resultant. b. Draw the man s final displacement vectr. c. Measure the length f the vectr n yur paper. cm (a) (c) d. What is this measurement in meters? m e. Use gemetry t determine the magnitude f the man s displacement. R R (b) (d) R R 5) man walks east fr 6 meters, then suth fr 8 meters, then west fr 12 meters. a. Sketch his path in the area belw using a scale f 1.0 centimeter = 2.0 meters. Vectr Cmpnents reak vectr int tw perpendicular cmpnents using X-Y system 6.0 m/s 30 Example v = 6.0 m/s at 30 Vectrs can be added tgether by adding their COMPONENTS Results are used t find: 3 P age

4 6) Calculate the x and y cmpnents f the fllwing vectrs. a. = 7 meters at 14 d. Sketch the resultant x and y vectrs n the axes belw. b. = 15 meters per secnd at 115 c. C = 17.5 meters per secnd 2 at 276 e. Calculate the length f the resultant generated by the resultant cmpnents. 7) dd vectrs D and F by fllwing the steps belw. a. Calculate the cmpnents f vectrs D and F. D = 35 meters at 25 F = 55 meters at 190 b. Calculate the sum f the x-cmpnents f vectrs D and F. c. Calculate the sum f the y-cmpnents f vectrs D and F.

5 Review Questins 1.2.1a Use a prtractr and ruler t add vectrs graphically. - Use a prtractr and ruler t cnstruct the resultant prduced when the tw vectrs in each system are added tgether Relative Mtin Examples train is mving east at 25 meters per secnd. man n the train gets up and walks tward the frnt f the train at a speed f 2.0 meters per secnd. What is his velcity? Relative t a fixed pint n the train m/s Relative t a fixed pint n the Earth m/s passenger n a 747 that is traveling east at 230 meters per secnd walks tward the lavatry at the rear f the airplane at a speed f 1.5 meters per secnd b Determine vectr cmpnents. - n bject mves with a velcity f 15 meters per secnd in a directin that is 25 east f nrth. If due east is 0, what is the reference angle f this vectr s velcity? [65 ] What is the passenger s velcity? What is the nrthward cmpnent f the bject s velcty? Relative t a fixed pint n the plane m/s [6.33 m/s] Relative t a fixed pint n the Earth m/s What is the eastward cmpnent f the bject s velcity? [13.6 m/s] 5 P age

6 Nn-Parallel Vectrs What happens t the verall velcity f the aircraft as the wind changes directins? v wind V thrust V thrust V thrust Practice 1) hckey player wh is 5.0 meters in frnt f a 1.2 meter wide gal slides puck directly at its center. The player releases the puck at 2.5 meters per secnd. s the puck slides tward the gal it drifts t the right at 0.4 meter per secnd. a. List the givens in the chart. With n wind, the speed depends nly n the aircraft thrust v wind b. Hw lng des it take fr the puck t reach the gal? v drift V thrust V thrust V thrust v wind v wind v wind c. Hw far des it drift in this time? v sht d. Is it a gal? Perpendicular Kinematics Critical variable in multi-dimensinal prblems is. swimmer mves acrss a 200 meter wide river at a velcity f 0.5 meter per secnd east. 200 m 2) ill is sitting n a tree branch that is 4.0 meters abve the grund with a bucket f water. He wants t dump the water n his friend Tim wh is appraching the tree at a cnstant speed f 8.0 meters per secnd. Hw far shuld Tim be frm the tree when ill dumps the water? Hw lng will it take the swimmer t get acrss? Nw assume that as the swimmer mves acrss the river, a current pushes him dwnstream (suth) at 0.1 meter per secnd. What des this mean fr the swimmer?

7 Review Questins 1.2.2a Determine the resultant speed/velcity in a tw dimensinal system. - girl rws a bat with a velcity f 2.5 meters per secnd due east acrss a river. s she rws acrss the river a current f 1.5 meters per secnd pushes her bat due nrth. Sketch a set f vectrs t represent these tw velcities and calculate the resultant speed f the bat. [2.9 m/s] Grund Launched Prjectiles Definitins prjectile is an bject that is mving thrugh the air with n means f prpulsin after it is launched. Can be launched with any at any May be subjected t What is the hrizntal part f the sccer ball s initial velcity? What is the vertical part f the sccer ball s initial velcity? 1.2.1b Determine an unknwn variable in a tw dimensinal system with cnstant velcity in bth directins. - plane flies nrth with a speed f 100 meters per secnd. t the same time, a crsswind pushes the plane east with a speed f 15 meters per secnd. In the time that the plane flies 300 meters nrth, hw far east will it drift? Hrizntal ssumptins 30 Vertical [ 45 m] - n ant crawls with a speed f 0.1 meter per secnd directly west acrss a cnveyr belt that is 0.5 meter wide. t the same time, the cnveyr belt mves suth. What speed was the cnveyr belt mving with if the ant mves 10 meters suth in the same time that it takes it t get acrss the belt? v i = a = max height v i = a = v f = [2.0 m/s] range 7 P age

8 Practice 1) baseball is hit with an initial velcity f 35 meters per secnd at an angle f 18 abve hrizntal. a. Sketch the baseball s trajectry. Indicate the pint at which the ball s vertical velcity is 0. 3) Tw tanks attack an enemy bunker n ppsite sides frm equal distance. Tank #1 fires its cannn at the bunker with an angle f 25 abve hrizntal. The rund leaves the gun f Tank#1 at velcity v and hits the bunker. (Ignre air resistance in this questin). Explain hw Tank #2 culd fire frm a different angle, but als hit the bunker. b. Calculate the vertical cmpnent f the ball s initial velcity. 4) Sketch a graph that shws the relatinship between the hrizntal range f a prjectile and the angle at which it is launched. c. Calculate the time that it will take fr the ball t reach the tp f its flight. Explain hw this amunt f time is related t the ttal amunt f time that the ball will remain in the air. Range d. Calculate the maximum height that the ball will reach Launch ngle ( ) 5) ball flies thrugh a parablic path and passes thrugh pints,, and C. C 2) place-kicker strikes a ftball with a hrizntal velcity f 22 meters per secnd and vertical velcity f 10 meters per secnd. The ball reaches its maximum height within 1.02 secnds. a. What is the ttal time that the ball will spend in the air? b. Calculate the hrizntal distance that the ftball will travel during its flight. Cmpare the a. vertical acceleratins,, and C. b. vertical velcities at,, and C. c. hrizntal acceleratins,, and C. d. hrizntal velcities at,, and C.

9 Review Questins 1.2.3a Determine cmpnents f a grund-launched prjectile s initial velcity. - prjectile is launched with a velcity f 200 meters per secnd at an angle f 35 abve the hrizntal. What is the vertical cmpnent f the prjectile s initial velcity? [115 m/s] What is the hrizntal cmpnent f the prjectile s initial velcity? [163 m/s] 1.2.3d Use equatins t determine: time f flight; max height; hrizntal distance; and/r launch angle fr a grund-launched prjectile. - ball is thrwn with an initial velcity f 15 meters per secnd at an angle f 20 abve the hrizntal. Determine the vertical cmpnent f the ball s initial velcity. [5.13 m/s] Determine the ball s ttal time f flight. [1.04 s] 1.2.3b Explain the assumptins needed t slve a grund-launched prblem. - What can yu assumer abut the hrizntal and vertical acceleratins f grund-launched prjectiles? [a hriz = 0; a vert = m/s 2 ] - What can yu assume abut the relatinship between the time that it takes fr a grund-launched prjectile t reach its maximum height and its ttal time f flight? 1.2.3c Sketch vectrs fr velcity r acceleratin at different psitns in a prjectile s path. Cmpare velcities and acceleratins at different pints. [t tt = 2t 1/2] - t each psitin:,, and C sketch a vectr t represent the vertical and hrizntal velcities f the bject. - Cmpare the vertical speed at t the vertical speed at. - Cmpare the hrizntal speed at t the hrizntal speed at C. - Cmpare the vertical acceleratin at t the vertical acceleratin at C. - Cmpare the hrizntal acceleratin at t the hrizntal acceleratin at. C - rck that is prjected with an initial velcity f 24 meters per secnd at an angle f 60 abve hrizntal takes 2.12 secnds t reach its maximum height. What is the hrizntal cmpnent f the ball s initial velcity? [12 m/s] What ttal hrizntal distance will the rck travel befre it lands? [50.88 m] - prjectile is fired with an initial vertical velcity f 12 meters per secnd and an initial hrizntal velcity f 15 meters per secnd. Determine the maximum vertical height that the prjectile will reach. [7.3 m] Determine the angle with which the prjectile was launched e Relate launch angle t: time f flight; max height; and hrizntal range. - Cmpare the time spent in the air fr a rck thrwn at an angle f 30 t the time spent in the air fr a rck thrwn at Cmpare the max height reached by a rck thrwn at an angle f 40 t the max height fr a rck thrwn at Cmpare the hrizntal range fr a rck is thrwn at an angle f 40 t the hrizntal range fr a rck thrwn at 75. [39 ] 9 P age

10 red ball rlls ff the edge f a table. What des its path lk like? Hrizntal Prjectiles Over the Edge s the red ball rlls ff the edge, a green ball is drpped frm rest frm the same height at the same time Examples n airplane making a supply drp t trps behind enemy lines is flying with a speed f 300 meters per secnd at an altitude f 3 kilmeters. Hw far frm the drp zne shuld the aircraft drp the supplies? Which ball hits the grund first? The red ball has an initial velcity. ut des nt have an initial velcity ssumptins The green ball falls frm rest and has n initial velcity stuntman jumps ff the edge f a 45 meter tall building t an air mattress that has been placed n the street belw at 15 meters frm the edge f the building. What minimum initial velcity des he need in rder t make it nt the air mattress? th balls begin with n th fall the same This means that the time f flight n astrnaut n the Mn thrws a wrench hrizntally with a speed f 0.5 meters per secnd frm a height f 1.5 meters. The astrnaut simultaneusly drps a feather frm the same height. Which bject will hit the grund first, the hammer r the feather? Why?

11 Review Questins 1.2.4a Explain the assumptins needed t slve a hrizntal prjectile prblem. - What can yu assumer abut the hrizntal and vertical acceleratins f hrizntally launched prjectiles? [a hriz = 0; a vert = m/s 2 ] 1.2.4c Use equatins t determine: time f flight; launch height; and/r hrizntal distance fr a hrizntally-launched prjectile. - n bject is thrwn hrizntally frm a height f 10 meters with an initial velcity f 8.0 meters per secnd. Determine the hrizntal distance that the bject will travel. - What critical assumptin must yu make abut the initial vertical velcity f a hrizntally launched prjectile? [v i = 0] 1.2.3b Sketch vectrs fr velcity r acceleratin at different psitns in a prjectile s path. Cmpare velcities and acceleratins at different pints. - t each psitin:,, and C sketch a vectr t represent the vertical and hrizntal velcities f the bject. - Cmpare the vertical speed at t the vertical speed at. - Cmpare the hrizntal speed at t the hrizntal speed at C. - Cmpare the vertical acceleratin at t the vertical acceleratin at C. - Cmpare the hrizntal acceleratin at t the hrizntal acceleratin at. - ball is thrwn hrizntally with an initial velcity f 5.0 meters per secnd. The ball travels 15 meters hrizntally befre it lands. What vertical height was it thrwn frm? [11 m] v [45 m] - prjectile is fired hrizntally frm a height f 15.0 meters abve the grund. The prjectile travels 40 meters hrizntally frm the pint frm which it was fired befre it lands. What initial velcity was the prjectile fired with? C [23 m/s] 11 P age