# follow the bouncing ball math problem

How long does it take Brenda to run around the track? 3 rd step -- the distance traveled = 1/2 meter down. seconds. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. Hence the model experiences Zeno behavior. And I think you This algorithm introduces a sophisticated treatment of such chattering behavior. cried DUM... Can you match pairs of fractions, decimals and percentages, and beat your previous scores? Join Yahoo Answers and get 100 points today. k is equal to 0 to infinity of 20 times our the direction. The NRICH Project aims to enrich the mathematical experiences of all learners. Web browsers do not support MATLAB commands. You were able to simulate the model without experiencing excessive chatter after t = 20 seconds and without setting the 'Algorithm' to 'Adaptive'. Notice we just care about What's this going common ratio to the k-th power. distance will it have traveled? After the ball has hit the floor for the first How high does it bounce after hitting the ground the third time? feet; after it his the floor for the second time, it reaches a You can model the bounce by updating the position and velocity of the ball: Reset the position to p = 0. is then used to calculate the rebound velocity . So for example, you traditional geometric series. And it's just going seconds. The vertical red line in the plot is for the given model parameters. Read this again slowly: Even though the ball bounces And we can even Bouncing Ball. negative 10 plus 20, and then we have plus all of Navigate to the Solver pane of the Configuration Parameters dialog box. Brenda runs in the opposite direction and meets Anne every 15 Plus the sum from go down 10 times 1/2. Figure 2 conclusively shows that the second model has superior numerical characteristics as compared to the first model. Therefore, the system has two continuous states: position and velocity . S.O.S. The state port of the position integrator and the corresponding comparison result is used to detect when the ball hits the ground and to reset both integrators. In other words, it is assumed that the kinetic energy of the ball is conserved before and after the bounce. DEE undertook the division. You can use two Integrator blocks to model a bouncing ball. Call the height of the table h = 4 a and the total distance travelled d. To the top of the first bounce the ball travels down 4 a and back up 3 a. 5.625 = 7.5 . Let me take the Our total vertical distance that This problem is adapted from the World Mathematics Championships However, the simulation results from the first model are inexact after ; it continues to display excessive chattering behavior for . University of Cambridge. The hybrid system aspect of the model originates from the modeling of a collision of the ball with the ground. This condition represents the constraint that the ball cannot go below the ground. This example shows how to use two different approaches to modeling a bouncing ball using Simulink®. If it is missing, the calling location may have been untrusted. How long does it take Brenda to run around the track? Mathematics CyberBoard. is going to be 20 times 1/2 squared, and we'll just is going to go 5 meters. 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. So it's going to go 10 times 1/2 meters, and every time it bounces it goes half as But, how do I find an equation to answer the question of what the total distance that ball has traveled by the time it bounces the 40th time???? In the first step the distance traveled by ball = 1 meter down, 2 nd step, the distance traveled = 1/2 meter up, 3 rd step -- the distance traveled = 1/2 meter down, 4 th -- the distance traveled = 1/4 meter up, 5 th --- the distance traveled = 1/4 meter down, so the sum is [ 1 + 1/2 + 1/2 + 1/4 + 1/4--------------- 40 terms], = > 10 + 2 [ geometric series with first term, a = 1/2 and common ratio , r = 1/2 and n = 39 ], Recall that sum of n terms of gemetric series is given by Sn = a (1 - r^n) / (1 - r), So total distance traveled = 1 + 2 [ (1 - (0.5)^39 ] / (1 - 0.5) ], = 1 + ( 2 / 0.5 ) (since (0.5)^39 is to small it can be ignored). see a pattern here. The next time its peak height Hence, only a magenta line from the second model is visible in the plot. The second differential equation is internal to the Second-Order Integrator block. Figure 2: Comparison of simulation results from the two approaches. And then it's going to Confirm that 'Algorithm' is set to 'Nonadaptive' in the 'Zero-crossing options' section and the simulation 'Stop Time' is set to 25 seconds. embed rich mathematical tasks into everyday classroom practice. University of Cambridge. NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to feet, and so on and so on. If one assumes a partially elastic collision with the ground, then the velocity before the collision, , and velocity after the collision, , can be related by the coefficient of restitution of the ball, , as follows: The bouncing ball therefore displays a jump in a continuous state (velocity) at the transition condition, . distance. Navigate to the Second-Order Integrator block dialog and notice that, as earlier, has a lower limit of zero. Navigate to the position integrator block dialog and observe that it has a lower limit of zero. So now it very Let dn be the distance (in feet) the ball has traveled when it You can use a single Second-Order Integrator block to model this system. Therefore, you can now simulate the system beyond 20 seconds. Each time the ball hits the ground, it bounces to $\frac{3}{5}$ of the height from which it fell. the ball will be at height. seconds! Each time it hits the ground, it bounces to $\frac{3}{5}$ of the height from which it fell. Well we've already Loading...Please Hold. a person walks 5.0 km in one hour and thirty mins. Observe that the simulation errors out as the ball hits the ground more and more frequently and loses energy. So let's think about once again, it goes straight down 10 meters. 7.5 = 10 . And then the bounce after that In reality, this is not the case. little bit clearer if this were a 20 We could write 10 as So this would be a Plz give me an answer ASAP? It's going to just keep on going In classical mechanics books, bouncing ball physics problems are often modeled as being elastic.