Phic - Projectile Motion A projectile i an object that fall through the air. Thee object are accelerated downward b the force of grait. The are alo affected b their paage through the air, to aring degree. We will ignore the effect of air reitance, howeer, which can be afel done if the object i dene and the ditance that it fall i not too great. Projectile alwa follow a cured path called a trajector (thi trajector i a egent of a parabola, a fact dicoered b Galileo). Soetie the trajector i called a ballitic path. Such otion i three dienional, but we will, for iplicit' ake, deal onl with otion in two dienion up/ down, and idewa. The ke to efficientl deal with projectile otion i to ipl break the elocit down into it horizontal and ertical coponent. Vector that are perpendicular to each other act independentl. Thi i o iportant that the Phic Kahuna will rewrite it in large friendl letter to drie hoe it iportance: Vector that are perpendicular to each other act independentl. Thi ean that the horizontal and ertical coponent of a elocit ector don't affect each other. The up and down otion ha nothing to do with the idewa otion and the idewa otion ha no effect on the up and down otion. Thi i like a reall iportant KEY CONCEPT! Auption: We are required to ake a couple of auption here: 1. g ha agnitude of 9.80 / and i alwa downward.. Effect of air reitance can be ignored. 3. Rotation of Earth can be ignored. 4. Motion (the horizontal elocit coponent) in the horizontal direction i contant.
The elocit of a projectile ha two coponent, and. = in = co Note that we hae alread atered the tak of calculating thee pek coponent. Projectile otion proble are quite iple the require no reall difficult atheatic nor do the reall ta our brain. With tated, none-the-le, for oe reaon, tudent often truggle with the. Since ou will not hae a whole lot of tie (we do got to oe fat in the old AP world), ou ut take the tie to ater the. The Phic Kahuna will help ou. In thi highl inforatie and uer-friendl pub ou will find in clear and concie proe all ou need to know. So what do ou need to know? Let find out. Iportant concept are: horizontal elocit coponent i alwa contant There i no acceleration in the horizontal direction. contant horizontal elocit ertical elocit increae with tie
Recall the Phic Kahuna deontration of the two bullet thing. The drawing aboe i a graphic depiction of the thing. The ball on the left ipl fall it ha no horizontal otion. The other ball i launched with a horizontal elocit of. Thi horizontal elocit doe not change and the ball oe idewa at a contant rate. The both fall downward at eactl the ae accelerating rate. The hit the ground at the ae tie. Do ou ee wh thi i o? A projectile launched upward at oe angle would hae a parabolic path that look like the drawing below. Drawn on the projectile i it elocit ector and the and elocit coponent. A long a the projectile i in the air, it will do two thing: It will oe horizontall at a contant peed. It will accelerate downward at a contant rate of g. The wa ou ole thee proble i to break it into two proble, a contant otion horizontal otion proble and a ertical contant acceleration proble. The bet wa to ee how to do thi i to jup in and ole oe proble. A flagpole ornaent fall off the top of a 5.0 flagpole. How long would it take to hit the ground? We aue that the ornaent ha no horizontal elocit. It fall traight down. We know how to do thi proble. 1 gt t g o that t 5.0 g t 9.80.6
A tone i thrown horizontall fro the top of a cliff that i 44.0 high. It ha a horizontal elocit of 15.0 /. We want to find how long it take the tone to fall to the deck and how far it will trael fro the bae of the cliff. Thi i like the flagpole proble, ecept that the tone ha an initial horizontal elocit. But we know that the tie it take to hit the ground i the ae a if it were falling traight down. (Thi i the ke concept!!!). So finding the tie i eactl like the preiou proble. Once we e found the tie, we can then find how far it trael horizontall. 1 gt 40.0 t g 9.80.86 Now that we know how long it take to fall, we can figure out the horizontal ditance it trael before it hit the ground. It ha a contant horizontal peed, and can trael idewa at thi peed a long a it i in the air falling, o to find we ue it aerage elocit and the tie: t 15.0.86 4.9 t A B-17 (a World War II era ultiengine bober) i fling at 375 k/h. The bob it drop trael a horizontal ditance of 5 50. What wa the altitude of the plane at the tie the had the old "bob awa"? We hae to find the tie for the bob to trael a horizontal ditance of 5 50 : Firt we conert the bober peed to eter per econd: k 1000 1 h 375 104. h 1k 3600 Then we can find the tie it take the bob to trael a horizontal ditance of 5 50. 1 t 5 50 t 104. 50.4 Now we can find the ertical ditance (altitude): 1 1 at 9.80 50.4 1 400
Upwardl Moing Projectile: 4.7 19.6 44.1 1 Projectile fall below traight line path One of the deontration that the Phic Kahuna i fond of i the hunter and the onke deo. Thi i the one where the onke i hanging fro a tree in the jungle and the hunter want to hoot it. Ecept that the onke can intantl detect a gun hot and will let go of the branch and fall traight down. So where, wa the ain idea, hould the hunter ai? Below the onke, at the onke, aboe the onke? 3 Monke and the Hunter Well we aw that the hunter hould ai traight at the onke. Thi i becaue the bullet will fall the ae ditance a the onke, o when ou ai at the onke and fire the round, the onke and the bullet will fall together and ou will end up drilling the poor innocent little critter. Phic can reall be cruel, can t it? The bullet wa actuall a banana
Here i the path of a projectile that i launched at oe angle to the horizon. It ha a horizontal elocit coponent and a ertical elocit coponent. For a long a it i in the air, it will be oing horizontall at. It will oe upward becaue of it initial ertical elocit,. Grait will act on it howeer, lowing it down. Eentuall, at the top of the path, it ertical elocit will = 0 - be zero. It will till hae it horizontal elocit coponent, howeer. Then it will begin to fall downward. When it finall reache the ae height it began with, it ertical peed will be the ae a what it began with, but the direction of it elocit will be downward intead of upward (a it wa at the beginning). The projectile will trael a horizontal ditance of (thi i often called the range). It will trael upward a ertical ditance of. In half the total tie of flight it will reach and it ertical elocit will be zero. There a lot of etr going on here. We alo aue that the projectile begin and end at the ae height. If thi i not true, it will be pelled out in the proble. A ball i gien an initial elocit of. 7 / at an angle of 66.0 to the horizontal. Find how high the ball will go. To ole thi proble, we hae to find the ertical elocit of the ball. Once we know it, we can find how high it goe. o in.7 in 66.0 0.7 The ball tart out with and rie till it ertical elocit i zero. We can ue thee a the initial and final elocit of the ball. o o o a 0 a a o a
0.7 9.80 1.9 Note that we hae to pa attention here to the ign of the otion. We hae both down and up otion and hae to be clear about which direction i poible. Of coure if the otion i in onl one direction, we don t hae to worr about it. A naal gun fire a projectile. The gun uzzle elocit (o peed of the bullet) i 345 / at an eleation of 3.0. What i the range of the hot? Firt find the ertical elocit: o in 345 in 3.0 18.8 at We know that o Now we can find the tie: o at t o g o t 18.8 18.8 9.80 37.31 Now we can find the range ince we know the tie. Firt we find the horizontal elocit: o co0 345 co3.0 9.6 t 9.6 37.31 10 900 t A punter ha a hang tie of 4.5. If the ball trael down the field 48, what wa the kick angle? To find the angle of the kick, we need to ue oe trig. Probabl the eaiet thing to do would be to find the horizontal and ertical coponent of the elocit and ue the tangent function to find the angle. The horizontal peed i ea to find - we know the tie and we know the range.
d 48 10.67 t 4.5 We can find the ertical elocit becaue we know the tie that the ball i in the air the tie to go up and the tie to go down. Let look at the path for the ball fro when it i at it a height to when it hit the ground. Thi ean that it initial elocit i zero and it final elocit i. o at at The tie for the ball to fall i half the total tie. 4.5 9.8.1.1 tan tan 64 10.67 1 1 o A tone i thrown off the top of a building fro a height of 45.0. The tone ha a launch angle of 6.5 and a peed of 31.5 /. (a) How long i the tone in flight, (b) how far fro the bae of the building doe it trael? (c) What i it peed jut before it hit the ground? (a) Firt we find it ertical elocit coponent. o in 31.5 in 6.5 7.9 45.0 Now we can ue thi to find the tie to reach it highet point. o 1 o at t 7.94.85 g 9.80 Now we can find how high it rie:
o o a 0 o a a 7.94 39.8 9.80 It took.85 to reach thi height, it will now fall thi ditance plu the height of the building before it hit the ground. So it will fall a ditance of: 39.8 45.0 84.8 We can find the tie to fall thi ditance: 1 84.8 at t 4.16 a 9.80 The total tie in the air i the total of thee two tie we found: 4.16.85 7.01 (b) We net find the ditance fro the bae. Thi i ea ince we know how long the projectile will be oing idewa: Firt we find the horizontal elocit: o co 31.5 co 6.5 14.55 t 14.55 7.01 10 t (c) To find the peed at the ground, we need to recobine the two coponent into the actual elocit ector. We can ue the Pthagorean theore for thi: Speed at ground: We need to find at the botto. We know that the rock fall for a tie of 4.16 fro it a height. We can ue thi to find it peed jut before it hit.
at 9.80 4.16 40.8 14.55 40.8 43.3 The firt digital coputer, called ENIAC (Electronic Nuerical Integrator and Coputer) becae operational in 1946. It wa funded b the Ar in World War II. The purpoe for the thing wa to calculate trajectorie for artiller hell to produce firing table that the gun crew who ered the gun could utilize. It could 500 nuber in onl one econd and could calculate the trajector for a firing proble in onl 30 econd a true iracle. It ued electronic tube and required 174 kilowatt of power (that 33 horepower for ou non-etric folk). Anwa, the power needed to ole one trajector proble wa about the ae a the aount of power generated b the powder charge during an actual fire iion for a ingle hell. Intereting. Here we go, one lat proble: You throw a potato at an angle of.. If the thing i in the air for 1.55, how far did it go, ditance-wie? In half the tie it will trael to it aiu height. It will fall back down to the earth in the other half of the tie, o we can look at half the path, a fro when it ha an initial elocit of to when it reache the highet point of it path where it ertical elocit i zero. 1.55 0 at at 9.8 7.60 Since we know the angle and the ertical elocit, we can find the horizontal 7.60 elocit. tan 18.6 tan tan. We can now find the ditance it trael ince we know the tie and the elocit: 18.6 1.55 8.9 t