# Flipping Coins

## Objective

Determine how "fat" a coin would have to be so that landing on its side is reasonable.

## Base Difficulty

Medium, because of difficulty of regression (line fitting)

## Concept

Clearly it is very hard, if not impossible, to land a coin on its side. This is due to the shape of a coin, and the concept of Center of Mass. See the picture below for a high speed photograph of a coin being flipped.

Even though the coin may hit its edge, it's essentially impossible for it to stop on its edge.

## Motivation

Now imagine that instead of a single coin, you had a whole roll of 50 coins, like the one below.

It would be almost impossible to throw up a whole roll of coins and not have it land on its side. The same is true for a pencil; however, for a can of soup it seems a lot more likely.

Somewhere between having 1 coin and having 50 coins, it seems like there should be a number which gives you about equal chance of landing your *coin* on its top or on its side.

It's **your** job to figure out how fat your *coin* needs to be.

## Materials

- 50 pennies (or dimes, nickels, quarters)
- Super Glue

# CAUTION

Super Glue is generally **toxic** which means it is not safe to inhale or ingest. Make sure you do the gluing in a well ventilated area, and that you wash your hands before eating.

## Hypothesis

How many pennies (or other coins) wide will your *coin* have to be before it lands on its side within the first 40 trials? Obviously you can choose any number between 1 and 50, since your *coin* must have at least 1 penny and you only brought 50 pennies with you.

## Procedure

- Begin your
*coin*with just a single penny. - Take another penny and Super Glue it to the
*coin*. - Wait for your
*coin*to dry - Flip (toss) your
*coin*40 times and record the number of times it lands on its side. - Repeat steps 2-4 until the
*coin*lands on its side every time.

It may help you to organize your data in a table:

Number of pennies | Number of side landings | |||

Trial 1 | Trial 2 | Trial 3 | Trial 4 | |

1 | ? | ? | ? | ? |

2 | ? | ? | ? | ? |

... | ... | ... | ... | ... |

50 | ? | ? | ? | ? |

You should repeat this experiment for at least 4 trials to fill your table and help guarantee accuracy of your results. To do this, flip each version of the *coin* 40·`n` times where `n` is 4 or more. After the first 40 flips, start counting for the second trial. After the next 40 start counting for the third trial, and so on. This way you don't need to waste more pennies than necessary.

## Analysis

Plot the data you acquired (using Excel plotting, perhaps) with the number of pennies in the *coin* on the x-axis and the number of side landings on the y-axis. You should have as many graphs as you have trials, although you could put all the plots on one graph if you use different colors.

Can you see any trends? Does the number of side landings change dramatically from one thickness to the next? Was your hypothesis correct?

## Extensions

- Take a look at the graphs that you have. Are they all similar? What did you think it would look like? Try and decide how many pennies give you about 32 side landings, and then 16, 8, 4, 2. You may not necessarily be able to find an integer penny value, but a decimal is okay too. Using these, try and figure out what number of pennies will likely give you one side landing out of 40. Can you determine from this what the probability of a single penny landing on its edge would be?
- Maybe you can try types of coins other than pennies. Don't waste too much money, but it's worth a shot to see if the mass of the
*coin*or the width of the*coin*makes a difference.