Dropping the Times Square Ball

It’s December 31st, and you’re getting ready to enjoy the new year celebration. This year your plans are different from the past. In 2023 you took an empty suitcase and walked around the block, in 2022 you ate 12 grapes, in 2021, you smashed a plate, in 2020 you banged on the wall with bread, and in 2019, you threw water out the window. Wanting to avoid the neighbors calling the cops again, you opt for something less noisy, and less physical. This year you plan to sit in front the TV and watch the Time’s Square Ball drop.

As you get your stuff together, you ponder on the Times Square, New Year's Eve ball. A sparkling master piece covered with lighted crystal panels and more majestic than a big orange or peach. A reminder of a tradition that extends long before Time Square with history rooted in the Greenwich time ball from 1833. Then the thought hits you, “...how does it hit bottom exactly as the countdown ends?” You assume that there must be some magic involved. It must be the work of elves!

Well, while “there may or may not be any elves involved, there is magic, the magic of math, the tool that underpins all of physics and engineering, and today, on the Eve of New Year’s Eve, ciaoMath! explores a bit of the math behind the Times Square ball drop. You can get more information on the Iconic Times Square Ball here: https://www.timessquarenyc.org/nye/nye-history-times-square-ball

NOTE: I will be using SI units; the math and physics work best in the meter-kilogram-second system. From the Times Square Ball website, we get ball weight, 5,386 kg, and diameter, 3.66 meters. It find out that it begins its descent precisely 60 seconds before midnight and travels a vertical distance of 43 meters to the bottom, where it stops at exactly 12:00 AM.

Using a simple model, the motion of the ball can be described using Newton’s equations of motion, assuming zero air resistance and a free-fall state (no mechanical systems moving the ball), with only gravity acting on the ball.  The acceleration due to gravity is approximately g = 9.8 m/s2 directed downward.

The first step is to write an algebraic equation to describe the linear displacement as the ball drops from rest:

Where:

• The function y(t) is the height y, of the ball at some time t seconds.

• y0 is the initial height of the ball.

• v0 is the initial velocity (which is zero since the ball starts from rest).

• a is the acceleration due to gravity. To represent gravity, we substitute g for a in the equation. 

Since the ball starts from rest, v0 = 0, so the equation simplifies to:

This equation tells us how the height of the ball changes over time as it falls. The negative sign indicates that the height decreases as the ball descends. To figure out how long it takes for the ball to reach the bottom, we re-arrange the equation to make t the subject of the formula:

Substituting known values into the above equation, y0 = 43 m and g = 9.8 m/s2.

Thus, the ball in free-fall will take about 3 seconds, much faster than it normally falls. So there has to be some sort of braking to slow the ball down (the mechanical system and air resistance). You could use the Drag Equation to figure out how much drag slows down the ball:

But then you will have to solve a differential equation, and you don’t have the time to do differential equations on New Year’s Eve. So, you think a bit and realize that there is no way air is slowing the ball down that much, so there has to be a mechanical system that does all the work. Since the ball must descend precisely 43 meters in 60 seconds, a carefully controlled mechanism must adjust the ball’s speed to maintain a constant average velocity.

To calculate the average velocity required for the ball to complete its descent in 60 seconds, we use the formula:

Where:

• d is the total distance traveled (43 meters),

• t is the time taken (60 seconds).

Thus, the ball descends at an average speed of approximately 0.72 meters per second. So, some mechanism is used to adjust the rate of fall. We will not get into the details of how they rigged up their mechanical systems as that is beyond the scope of this article.

So, while the ball’s descent is governed by the principles of gravity, there are carefully coordinated mechanisms and intricate timing involved to make it the amazing spectacle it is. In fact, the motors and lights are all computer controlled, and the exact timing is coordinated with GPS, which is satellite based.

While we simplified a lot to give a quick peek into the math involved in a ball drop, we also laid the groundwork for future installments namely, Newton’s equations of motion, air-resistance & drag and GPS. Simple algebra is a foundational element of describing motion, and it will be an important component in space, rockets and communications which we will explore later on.