Exploring the Effect of Initial Drop Height on the Ratio Between Successive Bounce Heights

Exploring the effect of initial drop height on the ratio between successive bounce heights of a bouncy ball

Introduction

It is incredible that physics can describe the behavior of all the objects in our world. The complex motion of satellites in orbit, the movement of charges in an electric field, and the change in potential energy on a roller coaster appear difficult and complicated to describe. Yet, they can be predicted by relatively simple physics equations, which increase our understanding of the world around us. I was interested in finding out more about objects I see on an everyday basis.

For my physics investigation, I wanted to focus on toys I had enjoyed as a child. I decided to use a collection I had at home: colorful rubber balls (nicknamed “crazy” balls). I always enjoyed collecting these bouncy rubber balls from arcade games as a child. At restaurants and party venues, there were often games with small prizes. I have fond memories of visiting my favorite pizza place, playing on flashing arcade machines, and bringing home a bouncy ball as my prize. I collected these bouncy balls throughout my childhood, and now have over 100 of all sizes and colors. I love all the different textures and patterns on the balls, and still look through my collection now and reminisce. I thought it would be nice to make use of my old collection, and incorporate bouncy balls into my physics investigation.

These balls have fascinated me since I was young; they rebound from almost every surface, and bounce much higher than other balls. These bouncy balls are the inspiration behind my investigation. There is a lot of research on the bounce heights of ordinary balls, such as ping-pong and tennis balls. However, there are few studies done on these bouncy balls since they are made of a different material. I decided to explore bouncy balls since they were unique compared to other studies. Though they are “crazy”, I want to see if I can describe their rate of kinetic energy loss with a simple exponential decay equation.

If these balls are released from a high drop height, they bounce uncontrollably, and are difficult to catch. Bouncy balls move both horizontally and vertically when they bounce, and have slight spin. The balls also have slight imperfections in their spherical shape. The motion of these bouncy balls interested me. I want to determine the effect of bounce number on rebound height, and if this is affected by drop height.

Exploration

Bouncy Balls

Bouncy balls are fun for children since they rebound at huge heights when bounced. This is because these balls are made from polyurethane and polyvinyl chloride, which gives the balls a higher coefficient of restitution (Q & A: Materials for Bouncy Balls). The coefficient of restitution (e) is defined as the relative speed after the collision divided by the relative speed before collision (Weisstein). If e is 1, the objects experience an elastic collision. Bouncy balls exert a strong force hitting the ground, so the ground also exerts a strong force on the ball. Most balls lose lots of energy when hitting the ground to friction and sound, but this does not happen as much with bouncy balls. I have always enjoyed playing with bouncy balls, but this fact fascinated me. Rubber bouncy balls are made from “long, tangled strings of carbon”, which rearrange shape when the ball hits the ground and exerts a larger force (Brown). I wanted to learn more about the bounce heights of bouncy balls, and the properties that made them especially “bouncy”.

Bounce Heights

Balls never experience elastic collisions; kinetic energy is converted to sound and heat when it makes contact with the surface, and the ball does not return to its initial drop height when dropped onto an even surface. The percentage of rebound, or “bounciness” varies with different balls. This motion can be described using an exponential decay curve, showing that this relationship can be described mathematically. I want to further investigate the ratio of successive bounce heights, and focus on bouncy rubber balls, as they are more interesting and have a personal connection to me.

Experimental Variables

The independent variable is the bounce number, and I will be measuring the first five bounces. This measurement will be in whole numbers with no uncertainty. The dependent variable is the rebound height, measured to the nearest centimeter. There will be measurement uncertainty, and I will take these measurements for five different initial drop heights, and see if this has an impact on the ratio of successive bounce heights. The bouncy ball used, the floor surface, the height measurement process, the location of the camera, and the camera used will all be kept constant. These controls eliminate some variability and hopefully create more accurate results. There were some uncontrolled factors, however, such as the exact location on the floor where the ball bounced. I could not control the horizontal motion of the bouncy ball, since it could have been released with a slight spin. The ball also could be dropped from slightly different locations and have a small initial push.

Manipulation of Variables

Independent Variable: Five different values for bounce number will be measured (1-5). The bounce number is simply measured by counting the bounces in the video.

Dependent Variable: The rebound height will be measured using the height measurement procedure detailed below.

Experimental Plan

Ball Selection: I originally wanted to compare different bouncy balls in my collection, since I had so many different sizes, masses, and textures. However, I realized that it would be difficult to compare different ball radii since every ball is made of a different material. The textures, weights, and sizes of the balls vary too much. I realized learning about the effect of bounce number on rebound height would be more testable.

Drop Height: I chose 5 different drop heights of 60, 50, 40, 30, and 20 centimeters. I made sure these heights were large enough to collect sufficient data, but small enough so that the ball’s motion was fairly controlled and it did not hit other objects. When I measure this height, the center of the ball will be at the measuring line. This will keep all measurements consistent and reduce variability.

Motion Detector

I originally planned to use a Vernier Motion Detector to accurately calculate the ball’s bounce heights. However, I quickly realized it would not work when I tried it. Firstly, the bouncy ball moved horizontally –out of the range of the motion sensor. The sensor was not able to measure the velocity since the ball moved out of the fixed area. The sensor also did not work well for determining the rebound height. It had to be connected to a LabQuest device, which needed to be calibrated carefully and often had inaccurate readings. Overall, I realized this would not be the best way to measure bounce height.

Height Measurement: I decided to release the balls from different drop heights and take a video of the bounces. I will create a large board with measurement lines so that the height can be seen clearly. The image on the left shows the measurements on the board I will use. I will pause the video (taken on iPhone 6) and see the exact rebound height for the first five bounces. I can use a slow motion video setting on the iPhone to slow down the video to record the height to the nearest centimeter. The camera (phone) will be fixed to the ground to keep the experiment controlled.

Bouncy balls are very uncontrollable and often bounce with considerable horizontal movement if dropped from a high distance since they often spin slightly in the air. Therefore, for this investigation, I will only look at vertical bounces. The balls need to be dropped from a fairly low drop height; otherwise, they will start hitting other objects and the data cannot be measured.

Experimental Method

The purpose of this investigation is to understand more about the bouncing of a crazy ball. Obviously, balls bounce less with each successive bounce, but I want to investigate what the exact ratio is between successive bounces, and if this is the same regardless of initial drop height. I am interested in understanding more about the rate of kinetic energy loss in the bouncy balls in my collection. I will use one bouncy ball and measure it at 5 different drop heights.

I plan to conduct 5 trials for each drop height, and record the bounce height for the first five bounces. The bounce height must be measured exactly when the ball reaches the peak of its rebound, when velocity is zero. I will average the rebound heights for each of the first five bounces. I will then take the data for each drop height and create a line of best fit for each of the five initial drop heights. I want to see if there is a similar ratio between successive bounces regardless of the drop height.

Materials

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·  Bouncy ball

·  Meter stick

·  Poster board with height measurements

·  Pencil

·  Sharpie

·  Tape

·  Video camera/recording device (iPhone 6)

·  Graphing software (Microsoft Excel)

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Analysis

Raw Data

These tables show the raw data from my experiment. The rebound height for each bounce was recorded from the video. This was measured to the nearest centimeter. At the bottom, the average rebound height for bounces 1-5 is recorded.

It is not possible for the bounces to be higher than the initial bounce height; this is due to parallax error that will be discussed further in the conclusion. The drop heights (60, 50, 40, 30, 20 centimeters) have an uncertainty of ± 0.5 cm. The results for all the bounce heights also have an uncertainty of ± 0.5 cm since they were measured to the nearest centimeter using the video. The height will be measured from the bottom of the chart, using the measuring board previously described.

Raw Data Tables: Bounce Height v. Bounce Number

60 cm drop / Bounce 1 / Bounce 2 / Bounce 3 / Bounce 4 / Bounce 5
Trial 1 / 63 / 56 / 50 / 45 / 39
Trial 2 / 71 / 62 / 54 / 45 / 38
Trial 3 / 65 / 55 / 46 / 37 / 31
Trial 4 / 68 / 57 / 48 / 41 / 39
Trial 5 / 66 / 54 / 47 / 40 / 35
Average rebound height (cm) / 66.6 / 56.8 / 49 / 41.6 / 36.4
50 cm drop / Bounce 1 / Bounce 2 / Bounce 3 / Bounce 4 / Bounce 5
Trial 1 / 54 / 46 / 39 / 33 / 28
Trial 2 / 50 / 44 / 38 / 35 / 30
Trial 3 / 48 / 43 / 37 / 32 / 29
Trial 4 / 48 / 43 / 37 / 33 / 28
Trial 5 / 55 / 47 / 43 / 37 / 33
Average rebound height (cm) / 51 / 44.6 / 38.8 / 34 / 29.6
40 cm drop / Bounce 1 / Bounce 2 / Bounce 3 / Bounce 4 / Bounce 5
Trial 1 / 50 / 42 / 35 / 28 / 24
Trial 2 / 49 / 42 / 35 / 30 / 26
Trial 3 / 46 / 37 / 32 / 27 / 23
Trial 4 / 47 / 43 / 35 / 29 / 24
Trial 5 / 50 / 42 / 34 / 27 / 23
Average rebound height (cm) / 48.4 / 41.2 / 34.2 / 28.2 / 24
30 cm drop / Bounce 1 / Bounce 2 / Bounce 3 / Bounce 4 / Bounce 5
Trial 1 / 34 / 28 / 24 / 22 / 18
Trial 2 / 34 / 28 / 25 / 22 / 17
Trial 3 / 33 / 29 / 26 / 21 / 17
Trial 4 / 34 / 29 / 24 / 22 / 19
Trial 5 / 36 / 32 / 27 / 23 / 20
Average rebound height (cm) / 34.2 / 29.2 / 25.2 / 22 / 18.2
20 cm drop / Bounce 1 / Bounce 2 / Bounce 3 / Bounce 4 / Bounce 5
Trial 1 / 19 / 17 / 16 / 14 / 11
Trial 2 / 21 / 18 / 15 / 12 / 10
Trial 3 / 22 / 18 / 15 / 13 / 11
Trial 4 / 23 / 19 / 16 / 14 / 11
Trial 5 / 24 / 20 / 17 / 14 / 12
Average rebound height (cm) / 21.8 / 18.4 / 15.8 / 13.4 / 11

Processing Data

Data was gathered and entered into an Excel spreadsheet, where it was then processed and used to create graphs and calculate uncertainties. The average of the bounce height for each of the five heights was taken.

Uncertainty Calculations

Absolute uncertainty was calculated by dividing the range of the trials (maximum minus minimum) by two. Fractional uncertainty was calculated by dividing the absolute uncertainty by the average bounce height. Percent uncertainty was calculated by multiplying the fractional uncertainty by 100.

Processed Data Tables: Average Bounce Height v. Bounce Number

Initial Drop Height: 60 ± 0.5 centimeters

Bounce Number / Average Bounce Height (± 0.5 cm) / Absolute Uncertainty (Dx); (range/2) / Fractional Uncertainty
(Dx)/x / Percent Uncertainty
(Dx)/x * 100 / X = (xavg + ∆x) cm
1 / 66.6 / 4 / 0.1 / 10% / 67 ± 4 cm
2 / 56.8 / 4 / 0.1 / 10% / 57 ± 4 cm
3 / 49.0 / 4 / 0.1 / 10% / 49 ± 4 cm
4 / 41.6 / 2.5 / 0.1 / 10% / 42 ± 2.5 cm
5 / 36.4 / 4 / 0.1 / 10% / 36 ± 4 cm

Initial Drop Height: 50 ± 0.5 centimeters

Bounce Number / Average Bounce Height (± 0.5 cm) / Absolute Uncertainty (Dx); (range/2) / Fractional Uncertainty
(Dx)/x / Percent Uncertainty
(Dx)/x * 100 / X = (xavg + ∆x) cm
1 / 51.0 / 2.5 / 0.05 / 5% / 51 ± 2.5 cm
2 / 44.6 / 2 / 0.05 / 5% / 45 ± 2 cm
3 / 38.8 / 3 / 0.1 / 10% / 39 ± 3 cm
4 / 34.0 / 2.5 / 0.1 / 10% / 34 ± 2.5 cm
5 / 29.6 / 2.5 / 0.1 / 10% / 30 ± 2.5 cm

Initial Drop Height: 40 ± 0.5 centimeters

Bounce Number / Average Bounce Height (± 0.5 cm) / Absolute Uncertainty (Dx); (range/2) / Fractional Uncertainty
(Dx)/x / Percent Uncertainty
(Dx)/x * 100 / X = (xavg + ∆x) cm
1 / 48.4 / 2 / 0.04 / 4% / 48 ± 2 cm
2 / 41.2 / 3 / 0.1 / 10% / 41 ± 3 cm
3 / 34.2 / 1.5 / 0.1 / 10% / 34 ± 1.5 cm
4 / 28.2 / 1.5 / 0.1 / 10% / 28 ± 1.5 cm
5 / 24.0 / 1.5 / 0.1 / 10% / 24 ± 1.5 cm

Initial Drop Height: 30 ± 0.5 centimeters