Pyic 0 Leon 13 Projectile Motion In Leon 1 to 9 we learned ow to decribe te otion of object wic were eiter oving orizontally or vertically. Now we wi to conider otion were te object i oving bot orizontally and vertically at te ae oent. I. Independence of vertical and orizontal otion (See page 10 to 11 in Pearon for an excellent dicuion of projectile otion.) In Leon 1, you learned about relative otion. You dicovered tat if a plane i pointed in a norterly direction and a cro-wind i blowing, te plane will end up flying on a coure tat i a cobination of te plane and te wind. It i iportant to note tat te plane and te wind are independent of eac oter, but teir cobined effect produce te final reult. plane wind reult For projectile otion, wen an object i trown up at an angle to te ground or projected orizontally off of a table or a cliff, te ae principle of cobined independent otion applie. However, in ti cae te vertical coponent of otion i being affected by te acceleration of gravity, wile te orizontal coponent i not affected by gravity. Te reult i tat te object follow a parabolic pat or trajectory. A parabolic pat reult becaue te projectile i falling due to te attraction of gravity and, at te ae tie, it i oving orizontally. In te trobocopic picture note tat te projected ball fall at te ae rate a a dropped object. Furter, te orizontal coponent of otion for te projected ball reain contant trougout te fall. Dr. Ron Lict 13-1 www.tructuredindependentlearning.co
For projectile te orizontal coponent of otion i not affected by gravity. Furter, for low oving object te effect of air reitance are negligible. Terefore, te orizontal coponent of otion i bet decribed in ter of contant orizontal velocity: dorizontal vorizontal t dorizontal vorizontalt Te vertical coponent of otion, on te oter and, i affected by gravity. Terefore, te vertical coponent of te otion i bet decribed by: d v t a t vertical 1 1 (were a = 9.81 / ) Exaple 1 A rock i trown orizontally fro a 100 ig cliff. It trike te ground 90 fro te bae of te cliff. At wat peed wa te rock trown? Te key to properly olve ti quetion i to realie tat ince te orizontal and vertical otion are independent of eac oter we olve te independently. Terefore we olve te proble a a vertical part and a orizontal part. vertical part It i iportant to note tat altoug te rock a orizontal velocity, it initial vertical velocity i zero. d = v 1 t + ½ a t v 1 = 0 (initial otion i orizontal only) a = -9.81 / d ( 100) t = 4.5 d = -100 a 9.81 / t =? orizontal part d x = 90 t = 4.5 v x =? v x dx 90 = 0 / t 4.5 Dr. Ron Lict 13 - www.tructuredindependentlearning.co
Wen an object i projected up fro te ground at oe angle () wit peed v o, oe of te peed i orizontal (v xo ) and oe of it i vertical (v yo ). Te key to olving ti proble i to calculate te vertical and orizontal coponent (v xo and v yo ) and ten treat te eparately. at axiu eigt v y = 0 v xo = v co v yo = v in Exaple A ball i projected at an angle of 37 o up fro te orizontal wit a peed of 0 /. A. Wat wa te axiu eigt? B. How long wa te ball in te air? C. Wat i te range? Δd ax 0 / v yo = 0 / in 37 37 o v yo = 1.04 / Δd x axiu eigt a = -9.81 / v 1 = 1.04 / v = 0 d ax =? v x = 0 / co 37 v x = 15.97 / v = v 1 + a d v v 1 0 (1.04 ) d ax a ( 9.81 ) = 7.4 tie in air v = 0 (at axiu eigt) tie up = tie down v v 0 1.04 1 t = 1.3 a 9.81 t = x 1.3 =.45 range v x = 15.97 / d x = v x t = 15.97 / (.45 ) = 39. Dr. Ron Lict 13-3 www.tructuredindependentlearning.co
II. Projectile proble Tere are a nuber of ting to keep in ind wen doing projectile proble: Since te orizontal and vertical coponent of otion are independent, projectile proble are olved a a vertical part and a orizontal part. If a velocity i given a an angle above or below te orizontal, one ut break up te velocity into orizontal and vertical coponent. Te tie tat te projectile i in te air i liited by ow long it take for gravity to pull te projectile to te ground. Terefore, te vertical part of te proble i often ued to deterine te tie. Wen a proble ak for te range it i aking for te orizontal ditance. Exaple 3 A rock i trown wit a peed of 30.0 / at 30 o below te orizontal fro a 150 eer cliff. How far away fro te bae of te cliff did te rock land? 150 v xo = v co v yo = v in v = 30 / co 30 v = 6.0 / 30 o 30.0 / v 1 = 30 / in 30 v 1 = 15.0 / tie in air Δd range v 1 = 15.0 / a = 9.81 / d = 150 v =? t =? v v ad 1 v v ad 1 v ( 15.0 ) ( 9.81 )( 150) v 56.3 d v t d 6.0 (4.1) d 109 v v1 t a 56.3 15.0 t 9.81 t 4.1 Dr. Ron Lict 13-4 www.tructuredindependentlearning.co
III. Practice Proble 1. A rock i trown orizontally at 10.0 / off of a 150 eer cliff. Wat i te range? (55.3 ). A quarterback trow a football at 15 / at 40 o to te orizontal. Wat i te axiu eigt of te ball? Wat i te range? (4.74,.6 ) Dr. Ron Lict 13-5 www.tructuredindependentlearning.co
IV. Hand-in Aignent 1. A peron fire a bullet orizontally fro a gun at te ae intant tat anoter bullet i dropped fro te ae eigt a te rifle barrel. Dicu wic bullet will it te ground firt.. One day in te jungle of te Aazon, profeor Glick wa talking a very rare pecie of onkey, projectilu avoidicu. He potted one up in a tree and decided to oot it wit a dart rifle. If te onkey begin to fall at te ae intant tat te dart leave te barrel of te rifle, explain wy te profeor ould ai directly at te onkey. 3. A an i tanding on te edge of te roof of a building tat i 88.1 ig. If e trow an object orizontally fro te roof at 1 /, ow far fro te bae of te building doe te object land? (156 ) 4. A an tanding at te edge of a cliff trow a tone orizontally at 40 /. Te tone land 360 fro te bae of te cliff. How ig i te cliff? (397 ) 5. A baeball i trown at 30 o above te orizontal wit a peed of 49.0 /. A. How long doe it take for te ball to reac it axiu eigt? (.50 ) B. Wat i te axiu eigt? (30.6 ) C. How long i te baeball in te air? (5.00 ) D. Wat i te axiu range of te baeball? (1 ) 6. If a tone i trown at 60 o above te orizontal wit a peed of 196 /, wat i te axiu eigt and te range? (1469, 3391 ) 7. An airplane i flying level at 80.0 above te ground wit a peed of 350 k/. Te bobardier wie to drop food and edical upplie to it a target on te ground. At wat orizontal ditance fro te target ould te bobardier releae te upplie? (393 ) 8. A ball i trown orizontally fro a window at 10 / and it te ground 5.0 later. Wat i te eigt of te window and ow far fro te bae of te building doe te ball firt it? (1. x 10, 50 ) 9. An artillery gun i fired o tat te ell a a vertical coponent of velocity of 10 / and a orizontal coponent of 360 /. If te target i at te ae level a te gun (a) ow long will te ell tay in te air and (b) ow far down-range will te ell it te target? (4.9, 15.4 k) *10. A girl tanding on te top of a roof trow a rock at 30 / at an angle of 30 o below te orizontal. If te roof i 50 ig, ow far fro te bae of te building will te rock land? (5.3 ) *11. A cannon i fired at 30 o above te orizontal wit a velocity of 00 / fro te edge of a 15 ig cliff. Calculate were te cannonball land on te level plain below. (3.7 x 10 3 ) *1. A golfer it a golf ball wit a pitcing wedge fro a tee tat i 15 lower tan te green. If te ball leave te tee at 35 / at an angle of 40 o to te orizontal and it land in te ole on te green, wat i te orizontal ditance fro te golfer to te ole? (101.3 ) (For extra practice do te Practice Proble on page 107, 109 and 111 in Pearon.) Dr. Ron Lict 13-6 www.tructuredindependentlearning.co
Building a Projectile Macine In ti activity you and your partner (axiu of two people) are aked to build a projectile acine tat i deigned to do one of te two tak below: 1. A orizontal projectile launcer. Te projectile launcer will be placed on a table in roo 05. Te projectile ut it te bull' eye target painted on te floor.. A parabolic projectile launcer. Te projectile launcer will be placed in a rectangle painted on te floor. Te projectile ay be launced at any angle you deire. Te projectile ut it te bull' eye target painted on te floor. Te following rule of deign and contruction ut be followed: Te launcer ut ave a triggering ecani tat old te launcer in firing poition until releaed. No exploive or flaable gae or liquid ay be ued. Any kind of non-letal projectile ay be ued. Te launcer ut be deigned and contructed by your group. Ue of ready ade launcer like ling-ot, air rifle, etc. are not peritted. Any kind of launcer device ay be contructed uing readily available aterial. Your launcer will be required to launc tree projectile in turn. If you ave any quetion, jut ak. You ay tet your launcer on ite any tie prior to te actual copletion date. Evaluation of your device i baed on te following criteria: Functionality doe it work a a launcer? Accuracy/ditance did you it te target? Dr. Ron Lict 13-7 www.tructuredindependentlearning.co