Hello, physics friends.

So I was watching the COPA America earlier this year, and I got talking to a friend about Leonil Messi's height.

He's 5 foot 6, which seems shorter than average.

Maybe?

Well, it turns out that the average height of an Argentinian man is 5 foot 8 and 1/2.

So is Leonil Messi's stature an advantage for him?

Well, the height of an average male professional soccer player is 5 foot 11.

So is height an advantage in sports?

Obviously, some sports attract tall people.

Michael Phelps is 6 foot 4, but Gabby Douglas is 5 foot 2.

Just last week, Olympic gymnast Simone Biles and Olympic volleyball player David Lee posted this photo on Twitter.

And these heights are pretty representative of their sports.

In swimming, 5 foot 9 for female Olympic gold medalists and 6 foot 3 for men.

That's around 7 and 9 inches taller than their respective averages, worldwide.

The average gold medal gymnast, on the other hand, was 5 foot 1 for women and 5 foot 4 and 1/2 for men.

Both an inch and a half shorter than their respective averages.

So swimmers are tall and gymnasts are short.

Volleyball players are tall and divers are short.

And then there's soccer players that are somewhere in the middle.

This seems like an obvious question, but is height an advantage in sports?

The answer seems pretty simple at first.

Volleyball players that are tall can spike more easily.

Short gymnasts have a lower center of gravity so they can balance on the balance beam.

You know, like, it's much easier to stack two books horizontally, so their center of gravity is low, than to stack them up end to end.

Try it.

And shorter people, on average, weigh less just because of scaling.

So they have less inertia.

That is, less tendency to keep going in the same direction.

And you can change direction and flip more quickly.

The same is true of diving.

So divers tend to be short.

It's less obvious what-- somebody's going to get hurt.

As I was saying, it's less obvious why swimmers are tall, though.

A swimmer's goal is to be as fast as possible in water while fighting friction and drag, 1,000 times more of it than in air.

So to accelerate quickly, a swimmer needs to apply a lot of force with her torso and limbs.

So we need to figure out how a swimmer's height affect her maximum possible force.

So the amount of force you can apply is proportional to the physiological cross-sectional area of your muscles, which indicates the number of muscle fibers contracting and releasing in that muscle.

So imagine we have two people-- one 6 foot tall person we're going to call Flow and one 5 foot tall person we'll call Bubbles-- with identical proportions.

Every part of Flow is 6/5-- or 1.2-- times bigger than Bubbles.

That means that if the radius of Bubble's bicep equals 3 inches, Flow's bicep radius would be 3 times 1.2, or 3.6 inches.

But the cross sectional area of the bicep is proportional to radius squared, because the area of any shape is approximately proportional to r squared.

So in this super overgeneralized case, Flow's bicep has 1.2 squared or 1.44 times more area than Bubble's.

So Flow can apply 1.44 times as much force, just by being 1.2 times taller.

And on top of that, being heavier won't really bog you down in water because of the buoyant force.

So tall people get a disproportionately large boon in the strength department just by being taller.

But gymnasts need to be strong, too.

So shouldn't they be tall as well?

Well, the type of strength you need to pull yourself through water is different than the type of strength needed for gymnastics.

In gymnastics, what matters is how your weight compares to your strength.

That is, your strength to mass ratio.

It's easier to hoist up a smaller mass.

So the ideal would be to keep the mass small while increasing strength.

Let's go back to Flow and Bubbles, although the names don't make quite as much sense now.

Flow is actually at a disadvantage, because tall people disproportionately gain mass.

That's because mass is proportional to volume, which is proportional to length cubed.

So with height, you're endowed with brute strength.

But it's not as easy to hoist and throw your body about.

That's why sprinters are, on average, taller than long distance runners, because they need that power, the big muscle strength, in order to increase their speed really quickly.

Whereas long distance runners need to maintain a pace for a long time, which is easier if you're lighter.

So now in soccer, the average female Olympic player is around 5 foot 6.

And the average male player is around 5' 11", closer to average height.

Well, soccer requires you to be more well-rounded-- fast and agile with the ball, which is better if you're light and short, but power to sprint and get those headers and put pressure on the other team.

So you need height there.

So statistically, it's better to be somewhere in the middle.

But, of course, it all depends on what position you play.

Now just because I'm tall doesn't automatically mean that I'm a good swimmer.

And just because my sister is short doesn't automatically mean that she is a good gymnast, although she actually is.

But a little math and physics can help explain how your height might give you a statistical, but not necessarily practical, edge in certain sports.

But there's an obvious caveat.

Short people and tall people aren't necessarily proportional.

A tall person might have disproportionately broad shoulders or long limbs.

But this process gives a ballpark mathematical relationship between height and physical strength.

There's actually an entire field of study called allometry, or scaling, devoted to figuring out how the physical abilities and characteristics of living things change depending on how big they are.

It's an interesting blend of biology, statistics, and physics.

There's actually a bunch of sports researchers in this field.

One researcher actually predicted how fast competitive rowers could go based only on their sizes and the weights of their boats.

And was accurate to within 1%.

Medical researchers who develop pharmaceutical pills use mathematical relationships by first giving medicine to mice, then using allometry to figure out what dosage a human, who might be 15 times taller than a mouse is long, might need.

Kind of strange, isn't it?

That you can make all these predictions just by knowing one thing.

That is, how tall something is.

That's one of my favorite things about science, is that one thing can be related to unexpected phenomena.

And you can describe it all using math.

Very cool.

Thank you so much for watching this video.

And happy physics-ing.