AP Physics - Free Fall

Transcription

1 AP Phyic - Free Fall Aritotle (384 3 BC), one of your baic ancient Greek philoopher, aid that thing fall becaue they want to regain their natural tate - earth with earth, water with water, and o on. Thu a rock will fall back to the earth to be with the other rock. Since a big rock poee more "earth", it will fall fater than would, ay, a feather (which i woefully inadequate in the earth amount category compared with your baic rock). Aritotle idea appear to be true becaue a rock certainly fall fater than a feather. In fact it made o much ene, that Aritotle' idea on the ubject were the accepted truth for around 000 year until the Renaiance. The firt cientific tudy of gravity wa done by Galileo Galilei ( ). He wa trained a a mathematician and wa a univerity profeor. In the late 1500' Galileo conducted a erie of experiment on gravity. He i uppoed to have demontrated that heavy object and light object fall at the ame peed. The act of doing experiment to find out what would happen thi wa a very daring idea. Here i Galileo' account of the experiment from hi book, Dialogue of two New Science. "But I, Simplicio, who have made the tet can aure you that a cannon ball weighing one or two hundred pound or even more, will not reach the ground by a much a a pan ahead of a muket ball weighing only half a pound, provided both are dropped from a height of 00 cubit...the larger outtrip the maller by two finger-breadth, that i, when the larger ha reached the ground, the other i hort of it by two finger-breadth. Galileo did not, a i popularly believed, tate that the object would hit the ground at the ame time he undertood air reitance. He did undertand that without air reitance, the object would fall at exactly the ame rate. Galileo wrote about doing the experiment a if he had done it everal time, but it i not clear where or when he did it. The tory that he dropped cannon ball from the Leaning Tower of Pia ha only one ource, hi lat pupil and biographer, Vincenzo Vivani. He decribe a very public event -- the entire univerity in attendance to witne the thing. But no one at the univerity ever mentioned witneing the event. So whether Galileo did or did not do the experiment i ort of up in the air. Galileo' idea that thing fall at the ame rate flie in the face of common ene. It eem reaonable that heavy thing ought to fall fater than light one To tudy gravity, Galileo found that he had to low it down. Thi wa becaue he couldn t meaure the time it took an object to fall with the crude intrument of the time. Gravity wa lowed down by having ball roll down inclined plane (ramp). Gravity till caued the motion, but it effect wa decreaed to the point where Galileo could gather ueful data. Galileo found that the ditance that accelerated object would travel wa proportional to the quare of the time. More on thi later. Acceleration of Gravity: On the earth, gravity exert a force on everything with ma. (A force i a puh or pull.) The force make all object accelerate downward, toward the center of the earth. Thi acceleration varie a tiny little bit depending on where you are - at the North Pole thi acceleration i m/ and at the Equator it ha a value of m/. Thi i becaue the earth i not a perfect phere. Fortunately we can afely ignore the tiny difference in the 6

2 acceleration of gravity. The value which i commonly ued for thi acceleration i 9.80 m/. In Englih unit it i 3.0 ft/. Gravity' acceleration i kind of pecial o it i given it very own little ymbol, g. g 9.80 m Drop a rock from the top of a cliff and, in one econd, it will reach a peed of 9.80 m/, after two econd it will be traveling at 19.6 m/, in three econd it going 9.4 m/, at four econd it peed will be up to 39. m/, and o on. It look like the rock will keep going fater and fater and fater until it mahe into the earth, and it would, if it were falling in a vacuum. The thing i, ee, that the air caue a frictional force that oppoe the rock' fall and low it down. For hort drop with dene object (like rock) we can reaonably ignore the effect of the air. Oh, the fancy, cientific term for thi force exerted by the air i drag or air reitance, ometime it i called wind reitance. At high velocitie or over long ditance, the drag can become ignificant, epecially for object that are not dene, like feather or leave or fluff. In the real world, an object in free fall will accelerate to it terminal velocity. Thi i the peed at which the force of gravity equal the drag force. The object then top accelerating and fall at a contant velocity. People jumping out of airplane experience thi. The typical laid out poition that ky diver ue give them a terminal velocity of around 100 mph. When an object i releaed and allowed to fall, it motion can be decribed by the following table (ignoring air reitance): Time Velocity Ditance m/ 4.6 m 19.6 m/ 19.6 m m/ 44.1 m m/ 78.4 m m/ 1 m &tc The kinematic acceleration equation can be ued to decribe the motion of falling object. A ball i thrown traight upward. If it take 4.5 econd to reach the top of it path, what i it initial peed? Since the ball i traveling upward, and the acceleration i downward, the ball will low down a it move up. For the upward part of it motion, it final velocity will be zero it will then momentarily come to ret and then change direction and begin to accelerate downward. Since we know that for the upward part of it journey the final velocity i zero, we can eaily calculate the initial velocity. 7

3 m m v v0 at v0 at The velocity and acceleration have oppoite direction, o one of the quantitie mut be negative. We ve choen down a the negative direction for the above olution (but hey, you could chooe up to be negative if you like). A tone i thrown traight up from top of building with an initial peed of 35.5 m/. (a) How high doe it go from the top of the building? (b) How much time to reach the maximum height? (c) If the building i 45. m tall, how much time will it take to hit the ground from when it wa initially launched? (a) o v v ay v at top i zero o; o 0v ay v 1 o m ay vo y m a m 9.80 (b) v vo at 0 vo m 1 t m 9.80 at t a v o (c) The tone take 3.6 to reach the highet point in it path, it then mut fall 64.3 m (to the top of the building) and then another 45. m to hit the deck below. So figure the problem from the top of the ball path, where it velocity i zero and jut before it begin to fall back down. It initial velocity i zero, and, ince the tone will be falling down, we can, what the heck, aume that down i poitive (we can do thi! We are in charge!): 1 y at t y a 64.3 m 45. m t m The total time for the ball to be in the air i

4 Negative or Poitive: You get to elect the coordinate ytem that you ue to olve problem. Thi mean you get to decide where the diplacement i zero and what direction will be poitive or negative. Look at what happen if you have a negative acceleration, uch a 9.8 m/. Doe thi mean the object i decelerating (lowing) or doe it mean that the object i moving along a negative (perhap the y) axi? It would depend on the problem. For an object moving on the x-axi it would mean decelerating. For an object falling along the y-axi, due to gravity, it mean the object i accelerating, but in the downward direction. You choe you your direction for thi tuff baically o that the calculation and everything are eaiet. Dear Doctor Science, Why doe a Kleenex, when dropped over a wate baket, alway end up on the floor intead of the bottom of the baket? -- Kevin Gutafon, M.D. from Minneapoli, MN Dr. Science repond: Kleenex i a brand name facial tiue, enjoying relatively high tatu compared to generic brand and lowly toilet paper. So the haughty tiue conider life in a wate baket, even after ue, repugnant. Thu, you have two option if you want to avoid picking up tiue from around the baket. Either purchae a lower grade of facial tiue, or get a nicer wate baket. I've een ome copper lined one with a tooled leather exterior that would attract even the top of the tiue pecking order. Dear Cecil: I it true cat alway land unharmed on their feet, no matter how far they fall? --A D DOO, via America Online Cecil replie: I love thi quetion. I love it becaue (1) it eem completely wild, () it nonethele appear to have ome cientific bai, (3) on examination the cientific bai i open to eriou quetion, and--thi i the bet part--(4) the Teeming Million figured thi all out by themelve. I may be able to retire from thi job yet. Here' the EP verion of the tory you heard, related to me by AOL uer Bmaffitt: "There wa a Dicovery Channel pecial on thi a while back. The truth i, after a few floor it doen't really matter [how far the cat fall], a long a the oxygen hold out. Cat have a nonfatal terminal velocity (ound like a contradiction in term, but mot mall animal have thi advantage). Once they orient themelve, they pread out like a parachute. There are cat on record that have fallen 0 torie or more without ill effect. A long a the cat doen't land on omething pointy, it' likely to walk away." You're thinking: no freaking way. But the believer trot out a 1987 tudy from the Journal of the American Veterinary Medical Aociation. Two vet examined 13 cae 9

5 of cat that had fallen out of high-rie window and were brought to the Animal Medical Center, a New York veterinary hopital, for treatment. On average the cat fell 5.5 torie, yet 90 percent urvived. (Many did uffer eriou injurie.) Well, we know cat have exceptional coordination and balance, o maybe that contributed to the high urvival rate. One cat, for example, i known to have urvived a 46-tory fall. (It apparently bounced off a canopy and into a planter.) But here' the weird part. When the vet analyzed the data they found that, a one would expect, the number of broken bone and other injurie increaed with the number of torie the cat had fallen--up to even torie. Above even torie, however, the number of injurie per cat harply declined. In other word, the farther the cat fell, the better it chance of ecaping eriou injury. The author explained thi eemingly miraculou reult by aying that after falling five torie or o the cat reached a terminal velocity--that i, maximum downward peed--of 60 mile per hour. Thereafter, they hypotheized, the cat relaxed and pread themelve out like flying quirrel, minimizing injurie. Thi peculation i now widely accepted a fact. But there' a potential fatal flaw in thi argument, which emerged from a dicuion on-- I can't uppre a grin--alt.fan.cecil-adam on the Uenet. (In fairne, the objection may have originally been raied on alt.folklore.urban.) The potential flaw i thi: the tudy wa baed only on cat that were brought into the hopital. Clearly dead cat, your baic fell-0-torie-and-look-like-it-came-out-of-acan-of-spam cat, go to the Dumpter, not the emergency room. Thi may kew the tatitic and make fall from great ditance look afer than they are. I called the Animal Medical Center to ee if thi poibility had been conidered. The original author were long gone, o I poke to Dr. Michael Garvey, head of the medical department and current expert on "high-rie yndrome." Dr. Garvey wa adamant that the omiion of nonreported fatalitie didn't kew the tatitic. He pointed out that cat that had fallen from great height typically had injurie uggeting they'd landed on their chet, which upport the "flying quirrel" hypothei. I uggeted thi merely meant that a cat landing in thi poition had a chance of urviving long enough to be brought into the hopital, wherea cat landing in other poition were o manifetly dead that the hopital wa never notified. Dr. Garvey didn't buy it, but aid thi wa a matter about which reaonable people might diagree. We await the formation of a committee of New York high-rie doormen to compile truly global tatitic on the fate of falling cat. Meanwhile don't believe omething jut becaue it wa on the Dicovery Channel, or for that matter in the Straight Dope. 30

6 Dear Cecil: Back when I wa a kid we ued to take the cat up on the roof and to it off. It wa jut a one-tory houe, o the cat didn't have far to fall. That little bugger would pread out hi arm and leg and glide on down, jut like a flying quirrel. He never eemed to mind it in the leat. He'd let u drag him up there again and again. It eem they have a natural ability to protect themelve from fall. Now that' cience! --Dave, via AOL Cecil replie: No, that' tupidity. I got another note telling about ome moron who dropped (a) a cat and (b) a chicken out of a Cena at 800 feet to ee what would happen. The cat urvived. The chicken didn't. While that might eem to validate the flying-quirrel hypothei, what it really tell me i that the teenage adit of the world have gotten the idea that cat are immortal, o anything goe. Nonene. Let' review the fact: 1. Nobody ay that cat will urvive any fall uninjured. Of the 13 cat brought to New York' Animal Medical Center after accidental fall, two-third required treatment, and half of that number required lifeaving treatment.. The flying-quirrel hypothei may well explain why ome cat urvive extremely long fall. No one ha demontrated that all cat will urvive long fall. On the contrary, from anecdotal account we know that at leat ome cat are killed--the death jut aren't reported. Cecil' aitant Little Ed got into a big online argument with a young fellow who wa enamored of the flying-quirrel hypothei. After Little Ed patiently explained the difference between ome and all, the young fellow conceded Cecil wa right to make point number two above. "But o what if Cecil wa right?" the young fellow aid by way of a parting hot. (I'm paraphraing here.) "Cecil' point wa boring. The flyingquirrel hypothei i intereting." OK, fine, it' intereting. The ditz pitching the kitty out of the Cena thought that wa intereting. Jut keep your hand off that cat. --CECIL ADAMS 31

7 Dear ir I write thi note to you to tell you of my plight, For at the time of writing it, I'm not a pretty ight, My body i all black & blue, my face a deathly grey, And I write thi note to ay why I am not at work today. While working on the 14th floor ome brick I had to clear, But toing them down from uch a height, wa not a good idea, The foreman wan't very pleaed, he i an awkward od, and he aid I had to cart them down the ladder in me hod. Well clearing all thee brick by hand, it wa o very low, So I hoited up a barrel and ecured a rope below. But in me hate to do the job, I wa too blind to ee, That a barrel full of building brick wa heavier than me. And o when I untied the rope, the barrel fell like lead, And clinging tightly to the rope, I tarted up intead. I hot up like a rocket, and to my dimay I found That halfway up I met the bloody barrel coming down. Well, the barrel broke me houlder a to the ground it ped, And when I reached the top, I banged the pulley with me head. But I clung on tightly, numb with hock, from thi almighty blow, While the barrel pilled out half it brick ome fourteen floor below. Now when thee brick had fallen from the barrel to the floor, I then outweighed the barrel & o tarted down once more. But I clung on tightly to the rope, me body wracked with pain, And halfway down I met the bloody barrel once again. The force of thi colliion halfway down the office block, Caued multiple abraion and a naty cae of hock, But I clung on tightly to the rope a I fell toward the ground, And I landed on the broken brick the barrel had cattered round. Well a I lay there on the floor I thought I'd paed the wort, But the barrel hit the pulley wheel & then the bottom burt. A hower of brick rained down on me; I didn't have a hope. A I lay there bleeding on the ground I let go the bloody rope. The barrel now being heavier, it tarted down once more. It landed right acro me a I lay there on the floor. It broke three rib and my left arm, and I can only ay, "I hope you'll undertand why I am not at work today." ---- anonymou 3