# Does Jewelry and Big Hair Slow Down Olympic Runners?

To be an Olympic champion, athletes spend years perfecting every aspect of their performance. That same attention to detail goes into the sleek, lightweight uniforms they wear, because at this level the tiniest of advantages can affect the outcome.

So what’s up with all the high-fashion hair and flashy accessories at the Paris Olympics? In the photo finish of the men’s 100-meter sprint, Noah Lyles of Team USA is seen rocking a hefty diamond-studded chain around his neck and a big Omega Speedmaster watch.

To be clear, these are not accessories *after* the fact—it’s not a case of athletes puttin’ on the ritz for post-race photos. This is all *during* the fact. They’re wearing this stuff in competition.

Lyles won his race, with a time 0.005 second faster than the other guy, but would he have posted a faster time without the extra weight? Could sprinter Sha’Carri Richardson have taken gold instead of silver in the women’s 100 meters without those long, flowing locks?

Personally, I’m still calling Richardson the winner for sheer jazz and pizzazz. But do these things make a difference in finishing times? This, friends, is a question for coach Isaac Newton.

## A Basic Running Model

If you really get into the biomechanics, the physics of running is quite complicated. But for our purpose, since we only want to estimate differences, a simple model will do just fine.

Right off the blocks, a runner gradually builds speed. However, even over a short distance like 100 meters, they don’t accelerate the whole time. At some point they reach a constant speed or even slow down a bit. I’m going to model a sprinter that accelerates during the first 30 meters and then hits a constant speed of 11 meters per second (25 mph). Plotting velocity as a function of time it looks like this:

,

Let’s focus on the acceleration phase of this sprint. If an object accelerates, there must be a *net force* acting on that object in the direction of the acceleration. That’s Newton’s second law: F_{net} = mass **×** acceleration. So what forces are there on a running human? Here’s a picture:

Vertically we have the downward pulling gravitational force (**F _{g}**), which is balanced by an upward-pushing “normal” force from the ground (

**F**). (Otherwise, you’d sink into the ground.) Horizontally, there’s a forward-pushing frictional force where the feet contact the track (

_{N}**F**), which is what makes the runner increase in speed, and finally there is a drag force (

_{friction}**F**), where the air pushes back against the runner. Now we can use this basic model to make some estimations.

_{drag}## Running With Bling

Suppose you want to wear your favorite jewelry. What would that do to your 100-meter sprint time? We can ignore air drag, since the jewelry wouldn’t affect it very much. And let’s say we have a runner with a mass (**m**) of 65 kilograms (143 pounds). In our model above, we assumed an acceleration of 2 meters per second per second (m/s^{2}). Let’s call this **a _{1}**. So we can write the forward-pushing force as:

,

What would happen if you add a gold chain weighing 10 grams? Your forward-pushing force (**F _{f}**) is the same, based on your strength, but you will have a larger mass to move (

**m**+

**m**, where

_{j}**m**is the mass of the jewelry). That means we must have a smaller acceleration (

_{j}**a**):

_{2}Using the first case to find the force, we get the following expression for the new acceleration.

You can see that adding weight will always reduce the acceleration. Plugging in the values for **m**, **m _{j}**, and

**a**we get a new acceleration (

_{1}**a**) of 1.99969 m/s

_{2}^{2}. That’s 99.98 percent of your original acceleration. Without the jewelry, you’d have a time of 11.8674 seconds. Adding the bling we get a slower time of 11.8683 seconds. But that’s a minuscule difference.

,

Hold on, though. That’s a thick chain Lyles is wearing, and the watch itself probably weighs some 150 grams. So let’s say the mass of the jewelry is 250 grams instead of 10. In this case we’d get an acceleration of 99.61 percent of our base model. Still a pretty tiny difference. If I was a running coach, I’d say that if the chain makes you feel good, go for it!

## Running With Hair

What about hair? Does it matter if you pull your hair back or let it stream out behind you? In this case, it’s not a case of extra mass but air drag. This is more complicated, but we can again get a rough estimate. One model for the drag force looks like this.

This drag force increases with the velocity (**v**), the density of the air (**ρ**), the cross-sectional area of the runner (**A**), and their shape (represented by a drag coefficient **C**). So as the runner ramps up their speed, the drag force also increases. Some people say that this is what makes the runner reach a constant speed. But that can’t be true—otherwise you could run infinitely fast on a treadmill, where there is no air drag.

So now let’s add this force to a runner during the acceleration phase of a 100-meter sprint. Although humans are a very complicated shape, we can get an estimate of the drag parameters by looking at a sky diver. When a human jumps out of an airplane, they increase in speed until the downward-pulling gravitational force (mass times the gravitational field) is equal to the upward-pushing air drag force. This is called terminal velocity, and for normal humans it has a value of about 120 miles per hour (54 meters per second).

If we take our 65-kg person, then the value of all the stuff in front of the velocity in the equation would be 0.218 Ns^{2}/m^{2}. (Just take my word for it and don’t worry about those complicated units.)

When I add this drag force into the model, I get a 100-meter time of 11.9884 seconds with a top speed of 10.6822 meters per second.

Now suppose you have very voluminous hair. In the extreme case, that might increase the cross-sectional area of the runner by 1 percent. That means the constant in front of the velocity would also increase by 1 percent. Great. That gives a run time of 11.9925 seconds. Oh snap. You just lost 0.0041 seconds off your time. Speaking as a guy with almost no hair to speak of, I’d say that’s totally worth it.