"Annenberg Media" "¶" "IT WAS DISCOVERED BY GALILEO," "REFINEDBYISAACNEWTON," "AND,INTHEHANDS OFALBERTEINSTEIN," "PROVIDEDA THEORY OF THE MECHANICS OF THE COSMOS." "ITWAS ONE OF THE DEEPEST MYSTERIES" "INALLOF PHYSICS." "ALLBODIESFALLWITH THE SAME CONSTANT ACCELERATION." "IN A VACUUM, ALL BODIES FALL" "WITH THE SAME..." "CONSTANT..." "ACCELERATION." "THAT'S THE LAW OF FALLING BODIES." "IT DOESN'T SEEM LIKE MUCH TO GET EXCITED ABOUT." "YET, JUST LOOK AT WHAT IT SAYS." "IT SAYS THAT THE EFFECT OF GRAVITY ON ALL BODIES" "IS THE SAME, REGARDLESS OF THEIR WEIGHT." "FROM GALILEO, TO ISAAC NEWTON RIGHT DOWN TO ALBERT EINSTEIN," "THAT'S BEEN ONE OF THE CENTRAL MYSTERIES IN ALL OF PHYSICS." "FURTHERMORE, IT SAYS BODIES FALL WITH CONSTANT ACCELERATION." "IT'S ALMOST IMPOSSIBLE EVEN TO UNDERSTAND WHAT THAT MEANS" "WITHOUT A MARVELOUS MATHEMATICAL DEVICE CALLED A DERIVATIVE." "WE'LL SEE TODAY WHAT THAT MEANS." "AND FINALLY, PROFOUND AND IMPORTANT THOUGH ALL THIS IS," "IT VIOLATES OUR SIMPLEST INTUITION" "BECAUSE IT HAPPENS ONLY IN A VACUUM," "NOT IN THE WORLD WE'RE FAMILIAR WITH." "FOR ALL OF US, THE EFFECT OF THE EARTH'S GRAVITY" "WAS PROBABLY OUR FIRST ENCOUNTER WITH THE LAWS OF NATURE." "WHETHER OR NOT WE UNDERSTAND HOW GRAVITY WORKS..." "WE HAVE AN INNATE FEAR OF ITS EFFECT." "BUT EXACTLY WHAT IS THE EFFECT OF GRAVITY?" "SOME BODIES FALL TO THE EARTH QUICKLY AND DIRECTLY," "WHILE OTHERS BEHAVE QUITE DIFFERENTLY." "IN SOME CASES," "GRAVITY CAN BE RESISTED ALMOST INDEFINITELY." "TO MAKE ANY SENSE ABOUT HOW AND WHY BODIES FALL," "WE MUST SEPARATE THE EFFECT OF GRAVITY ON A FALLING BODY" "FROM THE OPPOSING EFFECT OF THE AIR THROUGH WHICH IT FALLS." "IN OTHER WORDS, WE HAVE TO IMAGINE A BODY" "FALLING THROUGH A VACUUM." "IF A PENNY AND FEATHER" "DROP SIMULTANEOUSLY FROM THE SAME HEIGHT," "THEY BEHAVE AS WE'D EXPECT," "EACH FALLING AT A VERY DIFFERENT RATE." "BUT THAT'S ONLY BECAUSE OF THE EFFECT OF AIR RESISTANCE" "ON THE TWO OBJECTS." "IN A VACUUM, A PENNY, A FEATHER," "AND ANY OTHER OBJECT" "WILL FALL AT THE SAME RATE." "WITH VIRTUALLY NO AIR INSIDE THE GLASS TUBE," "THE PENNY AND FEATHER ARE NOW IN A VACUUM." "NOW WE'LL WITNESS THE LAW OF FALLING BODIES IN ACTION." "WITHOUT THE EFFECT OF AIR RESISTANCE," "ALL BODIES, REGARDLESS OF WEIGHT," "FALL AT EXACTLY THE SAME RATE." "WHEN APOLLO 15 ASTRONAUT DAVID SCOTT" "EXPLORED THE AIRLESS MOON," "HE COULDN'T RESIST REPEATING THIS CLASSIC EXPERIMENT." "[DAVID SCOTT] WELL,I HAVE AFEATHERANDAHAMMER." "ONEOFTHEREASONS WEGOTHERE" "WASBECAUSE AGENTLEMANNAMEDGALILEO" "MADEA RATHER SIGNIFICANTDISCOVERY" "ABOUTFALLINGOBJECTS INGRAVITYFIELDS." "WHERE WOULD BE A BETTER PLACE TOCONFIRMHISFINDINGS" "THANONTHEMOON?" "I'LLDROPTHEMHERE." "HOPEFULLY,THEY'LLHIT THEGROUNDSIMULTANEOUSLY." "HOWABOUTTHAT?" "ITSEEMSTHAT MR.GALILEOWASCORRECT." "MR. GALILEO WAS CORRECT." "NEARLY 400 YEARS AGO," "AT A TIME WHEN EVERYONE BELIEVED" "THAT HEAVY BODIES FALL FASTER THAN LIGHTER ONES," "GALILEO REALIZED THAT IN A VACUUM," "ALL BODIES SHOULD FALL AT THE SAME RATE." "GALILEO COULDN'T PRODUCE A VACUUM," "BUT HE COULD IMAGINE ONE." "HE PICTURED A HEAVY BODY ATTACHED TO A LIGHTER ONE." "WOULD THIS COMPOUND BODY FALL FASTER OR SLOWER" "THAN THE HEAVY BODY ALONE?" "IF THE LIGHTER BODY DID FALL MORE SLOWLY," "IT SHOULD SLOW DOWN THE HEAVY BODY." "BUT THE COMPOUND BODY IS ACTUALLY HEAVIER" "THAN THE HEAVY BODY ALONE." "THEREFORE, THE COMPOUND BODY SHOULD FALL FASTER" "THAN THE HEAVY BODY, NOT SLOWER." "OBVIOUSLY, THE VIEW THAT THE HEAVIER A BODY IS," "THE FASTER IT FALLS," "LEADS TO AN INESCAPABLE CONTRADICTION." "GALILEO REALIZED THAT THE ONLY LOGICALLY ACCEPTABLE VIEW" "WAS THAT ALL BODIES FALL AT EXACTLY THE SAME RATE" "ONCE AIR RESISTANCE IS REMOVED." "IF ALL BODIES IN A VACUUM" "FALL AT THE SAME RATE," "THE NEXT QUESTIONS IS, EXACTLY WHAT IS THAT RATE?" "FROM COMMON EXPERIENCE," "WE KNOW THE SPEED OF A FALLING BODY" "INCREASES AS IT FALLS," "WHICH MEANS THAT IT ACCELERATES," "DROPPING FASTER AND FASTER AS IT FALLS." "EVEN BEFORE GALILEO," "A NUMBER OF SCHOLARS TRIED TO FORMULATE" "A DESCRIPTION OF THIS ACCELERATED MOTION." "SOME 100 YEARS EARLIER," "LEONARDO DA VINCI MADE HIS OWN STUDY OF FALLING BODIES," "DRIVEN, PERHAPS, BY HIS DREAM OF HUMAN FLIGHT." "RATHER THAN ASK HOW FAST A BODY WAS FALLING," "DA VINCI ASKED HOW FAR WOULD IT FALL" "IN SUCCESSIVE INTERVALS OF TIME." "HIS THEORY OF ACCELERATED MOTION" "WAS THAT A BODY WOULD FALL GREATER DISTANCES" "IN LATER INTERVALS." "HE THEORIZED THAT THOSE DISTANCES" "WOULD FOLLOW THE INTEGERS." "1 UNIT OF DISTANCE IN THE FIRST TIME INTERVAL, 2 UNITS IN THE SECOND TIME INTERVAL, AND SO ON." "GALILEO ADOPTED DA VINCI'S METHOD OF DESCRIPTION," "BUT REACHED A DIFFERENT CONCLUSION" "ON HOW THE DISTANCE INCREASED." "GALILEO THEORIZED THE DISTANCES SHOULD FOLLOW THE ODD NUMBERS," "FALLING 1 UNIT OF DISTANCE IN THE FIRST TIME INTERVAL, 3 UNITS IN THE SECOND INTERVAL, AND SO ON." "IN OTHER WORDS, ACCORDING TO GALILEO, THE DISTANCE FALLEN" "IS PROPORTIONAL TO THE ODD NUMBERS." "GALILEO REACHED HIS CONCLUSIONS" "AFTER A BRILLIANT SERIES OF EXPERIMENTS" "IN WHICH HE TIMED A BALL" "AS IT ROLLED DOWN SUCCEEDINGLY STEEPER INCLINES," "MOVING CLOSER TO THE VERTICAL." "GALILEO'S LAW OF ODD NUMBERS" "CAN BE SEEN IN ACTION IN A VERY UNLIKELY PLACE," "WHICH WOULD HAVE AMAZED HIM" "EVEN MORE THAN THE SURFACE OF THE MOON" "AT MAGIC MOUNTAIN AMUSEMENT PARK IN SOUTHERN CALIFORNIA." "CUSTOMERS GLADLY PAY FOR THE PRIVILEGE" "OF PLUMMETING THROUGH SPACE UNDER THE INFLUENCE OF GRAVITY." "I'LL BUY YOU COTTON CANDY." "NO." "I DON'T WANT TO DO THIS." "ACTUALLY, THAT PART OF THE RIDE IS FREE." "HERE WE GO." "WHAT THE CUSTOMERS HAVE REALLY PAID FOR" "IS AN ARRANGEMENT THAT ALLOWS THEM TO SURVIVE." "AT ANY RATE, WHAT ABOUT GALILEO?" "IF THIS IS 1 UNIT OF DISTANCE," "THIS SHOULD BE 3," "THIS SHOULD BE 5, AND SO ON." "WHICH IS EXACTLY WHAT THEY ARE." "GALILEO WAS RIGHT." "IN SUCCESSIVE INTERVALS OF TIME," "THE DISTANCES FALLEN DO FOLLOW THE ODD NUMBERS." "THERE'S SOMETHING ELSE HERE THAT GALILEO UNDERSTOOD." "NOTICE THE TOTAL DISTANCE FALLEN AT EACH POINT." "AFTER THE FIRST TIME INTERVAL, 1 UNIT OF DISTANCE." "AFTER THE SECOND INTERVAL, 4 UNITS OF DISTANCE." "AFTER THE THIRD INTERVAL, 9 UNITS." "AFTER THE FOURTH, 16 UNITS." "IN OTHER WORDS, AT THE END OF EACH INTERVAL," "THE TOTAL DISTANCE FALLEN" "IS 1, 4, 9, 16, 25, AND SO ON." "AND THOSE NUMBERS, OF COURSE, ARE THE PERFECT SQUARES." "SO THE DISTANCE FALLEN" "IS PROPORTIONAL TO THE SQUARE OF TIME." "GALILEO'S LAW CAN BE WRITTEN AS A SIMPLE EQUATION," "USING s FOR DISTANCE" "AND t FOR TIME." "THIS MEANS WE'RE TALKING ABOUT DISTANCE" "AS A FUNCTION OF TIME." "THE DISTANCE s INCREASES AS THE SQUARE OF TIME." "THIS CONSTANT, c, IS NUMERICALLY EQUAL" "TO THE DISTANCE A BODY FALLS IN THE FIRST SECOND." "THAT'S 16 FEET, OR JUST UNDER 5 METERS." "AT ANY POINT IN THE FALL," "THE DISTANCE FALLEN IS EQUAL TO c TIMES THE SQUARE OF THE TIME." "SO AFTER 2 SECONDS," "THE DISTANCE FALLEN = c TIMES 2-SQUARED, OR 4c." "IF WE USE 16 FOR c," "WE KNOW THEY'VE FALLEN 64 FEET." "THIS SYMBOL EMPHASIZES THAT FOR ANY TIME t," "WE CAN FIND THE VALUE OF s." "AT THIS POINT, THE PETRIFIED FREEFALL RIDER" "CAN ASK HOW FAR SHE HAS FALLEN" "AT EACH INSTANT DURING THE PLUNGE." "THE MORE DISCERNING RIDER" "MAY ALSO WANT TO KNOW HOW FAST SHE'S FALLING." "IT'S THE DISTANCE SHE FALLS" "DIVIDED BY THE TIME IT TAKES." "SINCE SHE FALLS 64 FEET DURING THE FIRST 2 SECONDS," "HER AVERAGE SPEED MUST BE 32 FEET PER SECOND." "BUT THAT'S ONLY HER AVERAGE SPEED." "AT THE BEGINNING, SHE WAS STANDING STILL." "AT THE END OF TWO SECONDS," "SHE WAS FALLING MUCH FASTER THAN 32 FEET PER SECOND." "WHAT THIS WOMAN REALLY WANTS TO KNOW" "IS NOT HER AVERAGE SPEED," "BUT HER EXACT OR INSTANTANEOUS SPEED AT ANY GIVEN TIME." "HOWEVER, IF WE TRY TO USE THE SAME EQUATION," "DIVIDING THE CHANGE IN DISTANCE BY THE CHANGE IN TIME," "WE HAVE A SERIOUS PROBLEM." "AT ANY INSTANT DURING THE FALL," "LET'S SAY AT EXACTLY 1.5 SECONDS," "THE CHANGE IN DISTANCE AND TIME IS 0." "SO A FORMULA THAT DETERMINES SPEED" "BY DIVIDING THE CHANGE IN DISTANCE" "BETWEEN POINT A AND POINT B BY THE CHANGE IN TIME" "IS USELESS WITHOUT POINT B." "TO MAKE MATTERS WORSE," "THE TOP AND BOTTOM OF THE FRACTION WOULD BE 0." "DIVIDING BY 0 IS A MATHEMATICAL DISASTER." "PERHAPS THE EXPRESSION "INSTANTANEOUS SPEED"" "IS A CONTRADICTION IN TERMS." "YET, COMMON SENSE TELLS US THAT A MOVING OBJECT" "MUST HAVE A CERTAIN SPEED AT EVERY INSTANT." "THE PROBLEM IS MUCH MORE THAN A CLEVER PLAY ON WORDS." "IT'S A DILEMMA THAT PLAGUED MATHEMATICIANS" "FOR THOUSANDS OF YEARS." "THERE IS A WAY TO SOLVE IT." "INSTEAD OF ASKING THE INSTANTANEOUS SPEED" "AT AN EXACT TIME t," "ASK WHAT HER AVERAGE SPEED IS" "BETWEEN TIME t" "AND A POINT h SECONDS LATER" "AT TIME t + h." "THE CHANGE IN TIME IS h SECONDS." "IF THE DISTANCE FALLEN AT ANY TIME t" "= c TIMES t-SQUARED," "THE DISTANCE FALLEN AT TIME t + h" "MUST EQUAL c TIMES t + h-SQUARED." "THE PROBLEM IS SOLVED." "WE CAN CALCULATE HER AVERAGE SPEED" "STARTING AT ANY TIME t OVER ANY INTERVAL h." "h CAN BE 1 SECOND, 1/2 SECOND, 1/10 SECOND, OR EVEN 0," "BECAUSE NOW WE'RE NOT DIVIDING BY 0." "NOW WE CAN LET THE h INTERVAL" "SHRINK SMALLER AND SMALLER AND SMALLER," "EVEN TO THE ULTIMATE LIMIT." "AT THAT INSTANT, WE'VE CALCULATED A DERIVATIVE" "AS THE INTERVAL COMPLETELY SHRINKS TO 0." "IF h IS EXACTLY 0," "WE HAVE FOUND THAT AT ANY TIME t," "HER INSTANTANEOUS SPEED," "WHICH WE'LL CALL v, IS 2ct." "USING THE VALUE OF 16 FOR c," "WE CAN TELL HER, "MADAM, DON'T WORRY."" ""THE DISTANCE YOU'VE FALLEN IS 16 TIMES t-SQUARED FEET," ""AND YOUR SPEED AT EACH INSTANT" "IS SIMPLY 32 TIMES t FEET PER SECOND."" "OBVIOUSLY, SHE'S IMPRESSED." ""HOW DID YOU FIGURE THAT?" SHE MIGHT ASK." "WE JUST HAD TO INVENT THE DERIVATIVE." "IN COMMON USAGE, THE WORD DERIVATIVE MEANS ARISES FROM," "AS IN THE PHRASE, "FUDGE IS A DERIVATIVE OF CHOCOLATE."" "BUT IN MATHEMATICS," "THE WORD HAS AN EXACT TECHNICAL MEANING." "IT'S THE RATE AT WHICH SOMETHING IS CHANGING." "THE FALLING LADY'S SPEED WAS THE DERIVATIVE" "OF HER DISTANCE FROM THE TOP." "SPEED IS THE DERIVATIVE OF DISTANCE." "AT FIRST, WHEN WE DISCUSSED HER AVERAGE SPEED," "WE WERE DOING ALGEBRA," "SIMPLY PLUGGING NUMBERS INTO THE SPEED = DISTANCE DIVIDED BY TIME EQUATION." "BUT WHEN WE BEGAN WORKING WITH AN INTERVAL OF DURATION, h," "AND LET h SHRINK TO 0," "WE WERE CALCULATING A DERIVATIVE." "WE ENTERED THE WORLD OF DIFFERENTIAL CALCULUS." "DIFFERENTIAL CALCULUS" "IS THE MATHEMATICS OF USING DERIVATIVES." "CALCULATING A DERIVATIVE IS CALLED DIFFERENTIATION." "DERIVATIVES DON'T APPLY ONLY TO MOVING BODIES." "CONCEIVABLY, A DERIVATIVE COULD BE CALCULATED" "REPRESENTING THE RATE OF CHANGE" "IN DOLPHIN POPULATION VERSUS OCEAN TEMPERATURE," "OR THE VOLUME OF A BALLOON VERSUS ITS SURFACE AREA." "OR THE RATE OF CHANGE" "IN THE COST OF A PIZZA VERSUS ITS DIAMETER." "DERIVATIVES CAN BE CALCULATED FOR ALMOST ANY SITUATION" "IN WHICH ONE QUANTITY CHANGES" "AS ANOTHER QUANTITY INCREASES OR DECREASES." "TO GET FROM DISTANCE TO SPEED, WE CALCULATED A DERIVATIVE." "WHAT ABOUT THE ACCELERATION OF A FALLING BODY?" "TO GET FROM SPEED TO ACCELERATION," "WE DO THE SAME THING ALL OVER AGAIN." "IF v AS A FUNTION OF t = 2 ct," "THEN..." "BUT LOOK AT WHAT'S HAPPENED." "FIRST, THE DISTANCE s KEEPS INCREASING." "IT DEPENDS ON TIME." "IF t CHANGES, s CHANGES." "THE SPEED v ALSO KEEPS INCREASING WITH TIME." "BUT NOW WE'VE FOUND THAT THE ACCELERATION a" "DOESN'T DEPEND ON TIME AT ALL." "IT'S SIMPLY A CONSTANT." "a = 2c." "REGARDLESS OF THE VALUE OF t, a IS ALWAYS THE SAME." "WE'VE FINALLY DONE IT." "WE'VE FIGURED OUT THAT THE RESULT OF GRAVITY" "IS CONSTANT ACCELERATION." "WE HAD THREE QUESTIONS ABOUT A FALLING BODY" "HOW FAR, HOW FAST," "AND HOW FAST IS IT GETTING FASTER?" "HOW FAR WE FOUND OUT PRETTY EASILY" "BY WATCHING OUR FALLING LADY." "WE FOUND HER AVERAGE SPEED BY USING ALGEBRA." "BUT TO FIND OUT PRECISELY HOW FAST" "A BODY GOES AT EACH INSTANT," "AND HOW FAST IT GETS FASTER," "WE NEEDED OUR MARVELOUS NEW MATHEMATICAL TOOL," "THE DERIVATIVE." "USING THE DERIVATIVE, WE DESCRIBE FALLING MOTION." "BODIES FALL WITH CONSTANT ACCELERATION." "BECAUSE THAT ACCELERATION IS SO IMPORTANT," "IT HAS ITS OWN SYMBOL, A SMALL g." "AND g IS EQUAL TO 2c." "WE CAN PUT ALL THREE STATEMENTS" "OF THE LAW OF FALNG BODIES" "IN THEIR FINAL FORM" "BY REPLACING c WITH 1/2 g." "ACCORDING TO THE LAW OF FALLING BODIES," "A BODY FALLS WITH CONSTANT ACCELERATION," "WITH SPEED PROPORTIONAL TO TIME," "AND FALLS A DISTANCE" "PROPORTIONAL TO THE SQUARE OF TIME." "THAT KIND OF MOTION IS CALLED UNIFORMLY ACCELERATED MOTION." "IT IS DIFFICULT," "BUT NOT QUITE IMPOSSIBLE TO DISCOVER ALL THESE FACTS" "WITHOUT USING DIFFERENTIAL CALCULUS." "AND YET, GALILEO UNDERSTOOD ALL OF THESE FACTS." "IN FACT, NEARLY 300 YEARS BEFORE GALILEO," "A FRENCH SCHOLAR NAMED NICOLE ORESME" "HAD WORKED OUT THE BEHAVIOR" "OF UNIFORMLY ACCELERATED MOTION." "ORESME AND GALILEO USED NEARLY IDENTICAL MATHEMATICAL METHODS" "TO ANALYZE THE PROBLEM." "THEIR METHODS WERE BASED NOT ON ALGEBRAIC EQUATIONS," "BUT ON PROPORTIONS BETWEEN QUANTITIES" "AND ON GEOMETRIC FIGURES." "THE DERIVATIVE WAS INVENTED" "A GENERATION AFTER GALILEO'S DEATH" "BY SIR ISAAC NEWTON" "AND GOTTFRIED WILHELM VON LEIBNITZ." "WITH THIS POWERFUL NEW METHOD OF ANALYSIS," "EVEN MORE COMPLICATED KINDS OF MOTION" "COULD EASILY BE ANALYZED." "DESCRIBING UNIFORMLY ACCELERATED MOTION BECAME POSITIVELY SIMPLE." "WITHOUT DERIVATIVES, IT'S DIFFICULT TO UNDERSTAND" "WHAT ACCELERATION MEANS," "MUCH LESS DESCRIBE UNIFORMLY ACCELERATED MOTION" "AND WORK OUT ITS CONSEQUENCES." "YET, THAT'S WHAT ORESME AND GALILEO DID." "THEY DESCRIBED UNIFORMLY ACCELERATED MOTION" "AND WORKED OUT ITS CONSEQUENCES." "IT WAS SHEER GENIUS." "ONE OF THE JOBS OF PHYSICS" "IS TO FIND SIMPLE, ECONOMICAL, UNDERLYING PRINCIPLES" "TO EXPLAIN OUR COMPLICATED WORLD." "WE'VE DONE THAT." "IF I DROP SOMETHING," "IT FALLS UNDER THE INFLUENCE OF THE EARTH'S GRAVITY." "AS IT FALLS, ITS MOTION IS OPPOSED" "WITH VARYING DEGREES OF SUCCESS, BY THE AIR." "IF I IMAGINE DISPOSING OF THE AIR" "AND LETTING THE BODY FALL IN VACUUM," "THEN I DISCOVER A DRAMATIC AND SURPRISING FACT." "ALL BODIES FALL AT THE SAME RATE." "I COULD BE SATISFIED WITH THAT FACT." "DISCOVERING IT WAS AN IMPRESSIVE ACCOMPLISHMENT." "BUT WE'RE NOT SATISFIED." "WE WANT TO KNOW" "WHY IS IT TRUE?" "WHAT IS THE NATURE OF GRAVITY" "THAT LEADS TO SUCH STRANGE BEHAVIOR?" "THAT QUESTION IS ONE OF THE DEEPEST IN PHYSICS." "IT PERSISTED EVEN INTO OUR OWN CENTURY." "IT WAS THE STARTING POINT FROM WHICH ALBERT EINSTEIN" "BUILT HIS GENERAL THEORY OF RELATIVITY." "ONCE WE LEARNED THERE WAS ONE LAW FOR ALL FALLING BODIES," "THE JOB WAS TO EXPRESS THAT LAW WITH PRECISION." "ALL BODIES FALL WITH THE SAME CONSTANT ACCELERATION." "ACCELERATION IS THE RATE OF THE CHANGE OF SPEED." "AND SPEED IS THE RATE OF THE CHANGE OF DISTANCE." "WE HAVE THREE PRECISE MATHEMATICAL STATEMENTS" "OF THE LAW OF FALLING BODIES." "THEY ARE RELATED TO EACH OTHER" "BY ONE OF THE GREAT AND CRUCIAL DISCOVERIES" "IN THE HISTORY OF MATHEMATICS-- DIFFERENTIAL CALCULUS." "THE CALCULUS WAS DISCOVERED" "BY ISAAC NEWTON AND GOTTFRIED VON LEIBNITZ." "IT WAS A MIGHTY TRIUMPH," "THE MOST IMPORTANT EVENT IN MATHEMATICS" "IN THOUSANDS OF YEARS." "NEWTON AND VON LEIBNITZ" "SACRIFICED THE JOY OF THEIR DISCOVERY IN A BITTER DISPUTE" "OVER WHO DESERVED CREDIT FOR DISCOVERING IT FIRST." "NO, NO." "NO REPLY." "ACCORDINGTOTHELAW  OFFALLINGBODIES," "ABODYFALLS WITH CONSTANT ACCELERATION" "ATA SPEED PROPORTIONALTOTIME" "ANDFALLSADISTANCE" "PROPORTIONALTO THESQUAREOF TIME." "CAPTIONING PERFORMED BY THE NATIONAL CAPTIONING INSTITUTE, INC." "CAPTIONS COPYRIGHT 1985 CALIFORNIA INSTITUTE OF TECHNOLOGY" "THE CORPORATION FOR COMMUNITY COLLEGE TELEVISION THE ANNENBERG/CPB PROJECT" "PUBLIC PERFORMANCE OF CAPTIONS PROHIBITED WITHOUT PERMISSION OF NATIONAL CAPTIONING INSTITUTE" "Annenberg Media" "¶" "For information about this and other Annenberg Media programs call 1-800-LEARNER and visit us at"