Vold is a sadistic teacher who likes writing lots of exam questions. Find the sum of each infinite geometric series, if it exists. The bouncing ball geometric series is a nice example related to zenos paradoxes that forces students to think about how infinitely many discrete steps can sum to a finite answer. Also describes approaches to solving problems based on geometric sequences and series. In this video we use a geometric sequence to determine how high a ball is bouncing and an infinite geometric series to determine the total.
What is the formula for the bouncing ball problems. The geometric series is an important example of an infinite series. If a geometric series is infinite that is, endless and 1 1 or if r infinite series. Find the total vertical distance travelled by the ball. Our assumption means that even when the jumps are very tiny, the mechanism stays the same, and so the ball actually bounces infinitely many times. A ball starts its fall from a height h and then bounces back up to a heart rh where r is the coefficient of restitution. The second version also makes use of geometric series. Infinite bouncing ball while playing squash me and my friend started discussing something. Find the total distance the ball will travel if it is assumed to bounce infinitely often. To answer this question, it is important to get the frame of reference right.
Notice that the series ii is contained in both the downward and upward series, then s can be written as. As a final example we can describe how the geometric series can be applied to the problem of a bouncing ball. The solution is found by inserting information about r and the height from which the ball was dropped into a formula for an infinite geometric sum. The total distance s the ball travels can be found by adding the sums of these infinite series. Each time it strikes the ground it returns to threequarters of its previous height. You may come across this type of series when dealing with physical processes, such as the height of a bouncing ball, or in other areas of maths, such as fractal geometry. If you had a ball that was shot from the center of a square and started bouncing of the walls and hit a single real point every time the distance in the circumfrence of the square or whatever. In this article, the authors discuss some variations on this general idea, involving measurements of timesrather than heightsfor various combinations of bounces. Thus to find the total distance of travel we have to sum up a series. Author information robert styer is a faculty member in the department of mathematical sciences and morgan besson is a faculty member in the department of physics, both at villanova university.
This can only be summed if the progression is converging, which is true here as the ball after bouncing looses. So if we want to write the series as s20 12n as n goes from 0 to infinity, that series says the ball bounced up 10m and back down 10m, but that is not what happened, it only fell 10m, therefore we. Finding total distance and time for a bouncing ball application of series. Falling, rebounding, use the formula for an infinite geometric series with 1 dec 31, 2014 to answer this question, it is important to get the frame of reference right. The metaphysics of ifa ifa as an infinite geometric series.
To see this, compute and graph the sum of the first n terms for several values of n. He usually starts out the semester with only 10 questions on the first exam, but for each subsequent exam he writes one and a half as many questions as were on the previous exam. Geometric seriesbounce height of a ball mathematics. A ball is dropped from a height of 6 feet and begins bouncing. Geometric series word problem bouncing ball math help forum. References for more information about the physics of a tennis. That is because its motion is described by a series which though infinite, is convergent. Formulas for calculating the nth term, the sum of the first n terms, and the sum of an infinite number of terms are derived.
If we say that the tortoise has been given a 10 m head start, and that whilst the tortoise runs at 1 ms, achilles runs at 10 ms, we can try to calculate when achilles would catch the tortoise. Suppose you drop a basketball from a height of 10 feet. The height of each bounce is threefourths the height of the previous bounce. Exarnple 4 a bouncing ball loses half of its energy on each bounce. We modify a traditional bouncing ball activity for introducing exponential functions by modeling the time between bounces instead of the bounce heights. So, we dont deal with the common ratio greater than one for an infinite geometric series. Use the formula for an infinite geometric series to find each sum. In all workbooks reference books, when summing up the total distances of the paths taken by a bouncing ball, we first need to find the value of the distance when the ball bounces for the first time.
If this ball is dropped from a height of 9 ft, then approximately how far does it travel before coming to rest. Series solves for an unknown in the equation for the sum of a series of a geometric or arithmetic sequence. The total distance traveled by the ball is the sum of f and r, 30 ft. The falls form this geometric series with 10 terms, with first term 16, common ratio 12. Let d n be the distance in feet the ball has traveled when it hits the floor for the nth time, and let t n be the time in seconds it takes the ball to hit the floor for the nth time. If a ball is dropped from a height of 47m, and after each rebound, the height is 15 of its previous height, calculate the total vertical distance travelled. In this question, the ball always bounces back to a height which is a certain fraction of the ball s maximum height before the bounce. Use infinite geometric series as models of reallife situations, such as the distance traveled by a bouncing ball in example 4.
Except the ball only fell the first time, it did not bounce up and down like all the successive bounces do, so it only traveled the 10m once. One can analytically calculate the exact time when the ball settles down to the ground with zero velocity by summing the time required for each bounce. Notice that this problem actually involves two infinite geometric series. In my experiment, the ball was dropped from a height of 6 feet and begins bouncing. In the example of the bouncing ball dropped from a height of 9 feet and bouncing up two. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Even though this series has infinitely many terms, it has a finite sum. Whys the total distance travelled by a ball thats bouncing up and down modelled on the sum of an infinite geometric series when the ball will only bounce a finite number of times in reality.
Simulation of a bouncing ball matlab and simulink tutorials. You will use mathcad to calculate the sum of an infinite series. Modeling a bouncing ball with exponential functions and. You will use an infinite series to measure the total vertical distance traveled by the bouncing tennis ball. Zeno behavior is informally characterized by an infinite number of events occurring in.
The bouncing ball discussed in mathquestions and answers. The sum of an infinite geometric s for the series described above, the sum is s 1, as expected. The corresponding series can be written as the sum of the two infinite geometric series. Ball bounce a ball is dropped from a height of 10 feet. Falling, rebounding, use the formula for an infinite geometric series with 1 ball will travel approximately 168 inches before it. It does abnormal problems also included in the fst texbook i used bouncing ball problems and banking problems. A ball bounces 8 times find distance travelled geometric series. Mathematically we can express this idea as an infinite summation of the distances travelled each time.
Thus after each successive bounces, the distance becomes 0. Eric ej902125 exploring the mathematics of bouncing balls. Each time the ball bounces, it reaches 20% of its previous height. Bouncing ball geometric sequence question mathematics stack. But in the case of an infinite geometric series when the common ratio is greater than one, the terms in the sequence will get larger and larger and if you add the larger numbers, you wont get a final answer. A reboncing ball rebounces each time to a height equal to one half the height of the previous bounce. Purpose in this lab, you will analyze a bouncing tennis ball. Therefore we can use the infinite summation formula for a geometric series which was derived about 2000 years after zeno. This time is the sum of an infinite geometric series given by. Bouncing ball geometric sequence question mathematics. A bouncing ball is one of the simplest models that shows the zeno phenomenon.
A ball is dropped from a height of 40 feet, and each time it bounces. If anyone knows, please share, my math teacher is just torturing me with this problem. Numeric example in my experiment, the ball was dropped from a height of 6 feet and begins bouncing. Each time it hits the ground, it bounces to 80% of its previous height. A ball is dropped from a height of 1 meter onto the floor. Lastly, not only can the infinite geometric series of ifa system i. This lesson explains the good old bouncing ball problem. So if we want to write the series as s20 12n as n goes from 0 to infinity, that series says the ball. Using infinite geometric series consider the following infinite geometric series. Calculating the sum of the values in an infinite geometric series. Since no actual ball can do that, a real ball probably stops a little sooner. After the ball has hit the floor for the first time it rises 10.
Learn how this is possible and how we can tell whether a series converges and to what value. Geometric sequences and geometric series mathmaine. The mathematical ball bounces an infinite number of times. Geometric series word problem bouncing ball math help. It explains how to use geometric series to find the total distance of the bouncing ball. How can you find the sum of an infinite geometric series. You will use maple to calculate the sum of an infinite series.
Secondorder integrator model is the preferable approach to modeling bouncing ball. Its a bit of a curiosity that a bouncing ball actually does come to rest. If the common ratio is between 1 and 1 r infinite geometric series, the sum will converge to a finite sum. Each time it strikes the ground it bounces vertically to a height that is 34 of the preceding height. A ball dropped from a height of 12 feet begins to bounce. Bouncing ball suppose that a ball always rebounds 23 of the distance from which it falls. I know that the ball does stop bouncing and it is somehow related to geometric series and zeons paradox i think thats how you spell it. In this article, the authors discuss some variations on this general idea, involving measurements of timesrather than heightsfor various combinations of. Use the ideas of geometric series to solve the problem bouncing ball suppose that a ball always rebounds 23 of the distance from which it falls. Use the ideas of geometric series to solve the problem. We will also learn about taylor and maclaurin series, which are series that act as. How to recognize, create, and describe a geometric sequence also called a geometric progression using closed and recursive definitions. In this problem, we explore some applications of geometric series. One series involves the ball falling, while the other series involves the ball rebounding.
Oct 22, 2009 a ball is dropped from a height of 10 m. Watch sal determine the total vertical distance a bouncing ball moves using an infinite geometric series. To write a geometric series in summation notation, it is convenient to allow the index i to start at zero, so that a, a, a, ar, a, ar2, and so on. As a consequence, we can also model the total time of bouncing using an infinite geometric series. Using the geometric series in an applied setting 1. This is because r n approaches zero as n increases without bound. This problem is actually a sum of infinite terms of a geometric progression. Infinite series are sums of an infinite number of terms. Now i know how to find the answer of vertical distance using infinite geometric series.
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