# How Should You Launch a Ball to Achieve the Greatest Distance?

Physics gives a precise answer, at least in an ideal setting

In the Projectile Motion episode of NBC Learn's "The Science of NFL Football," you see that punted footballs travel in an arc known to mathematicians as a parabola.

In any football game both teams square off against each other and against a shared opponent as well—gravity. Earth's gravitational pull makes long-range passing a challenge and pulls down even the hardest-struck punts and placekicks.

Because gravity is a constant, experienced quarterbacks and kickers can account for its effects to move the ball downfield as efficiently as possible. Like all projectiles, a football, once released, follows a path known in mathematical terms as a parabola—a symmetric arc that eventually returns the ball back to the ground. (In real life a projectile's flight is affected not only by gravity but by wind and drag from air resistance, so the parabola would not be perfect.)

Parabolas have been studied for millennia, and their properties are well understood. For any projectile under gravity's influence, the distance attained during its flight is equal to sin(2θ) X v²/g, where v is the projectile's initial speed, g is the acceleration toward Earth due to gravity and θ is the angle at which the projectile is launched.

That may look like a complicated equation, but a couple of the variables can be ignored. First, because the force of gravity is constant, g will be the same no matter how a punter kicks the ball. Second, for a punter trying to boot a ball as far as possible, you can assume that he is kicking as hard as he physically can, so v depends simply on how hard he can kick, not on any strategic decision for a given punt.

The only choice he has to make to maximize distance, then, is the angle at which he kicks the ball. You can see from the equation above that the distance traveled by the ball will be greatest when sin(2θ) is greatest. The sine function reaches its largest output value, 1, with an input angle of 90 degrees, so we can see that for the longest-range punts 2θ = 90 degrees and, therefore, θ = 45 degrees. A projectile, in other words, travels the farthest when it is launched at an angle of 45 degrees.

But what about trying to maximize a projectile's height to increase hang time? In a parabola the peak height attained by a projectile is equal to (sin(θ))² X v²/2g. Once again, we can ignore v and g, for the same reasons as above. (Anyone looking to loft a projectile as high as possible would simply launch it as fast as possible, and gravity is constant.)

So to send a projectile flying as high as it can go, you can see that you want to make (sin(θ))² as large as possible, which simply means making sin(θ) as large as possible. As mentioned above, the sine function reaches its biggest output value, 1, with an input angle of 90 degrees, so we can see that for a sky-high punt θ = 90. That means that the best way to launch a high-altitude projectile is to send it flying at a 90-degree angle to the ground—straight up.

Of course, a vertical punt doesn't help much with field position, so you're not likely to see a 90-degree punt on the football field anytime soon. Not on purpose, anyway.

Rights & Permissions

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1. 1. Johnay 10:48 AM 11/9/10

So if the goal for a punter is to put the other team as far back as possible while giving their team ample time to move in for the tackle, given that the optimal outcome is to have the receiver downed right about at the goal line, the punter should kick their hardest at an angle calculated to have the ball land as close to the goal line as possible without going over.

2. 2. blah11235813 12:46 PM 11/9/10

The path of a punted ball is not a Parabola. It is in reality an Elliptic orbit.

3. 3. reflectogenesis 12:56 PM 11/9/10

I don't thinks this takes into account the biomechanics which dictate the point of contact which carries the maximum kinetic energy which can be imparted to the ball. This is computationally complex.
Perhaps depending upon things like temperature and possibly humidity. This might also affect the brain state affecting the synergy of different components of the process - creating a 'zone' in which fluid transmission of force through the contact with the ball results in optimal trajectory and perhaps rotational dynamics.
Reflectogenesis@hotmail.co.uk
Peter Reynolds

4. 4. Chris G in reply to blah11235813 01:52 PM 11/9/10

Well, if you want to that pedantic, neither is gravity a constant, and air resistance roughly goes up with the square of the velocity, but the density of the air is not a constant either, etc.

However, to a couple of orders of magnitude, and assuming the end zone is out of the kicker's range as was stated by the author, the model above is close enough.

One of my more interesting teachers had a way, as we progressed to higher levels of understanding, of saying, "Remember when I told you XYZ, well, that was a little lie. Here is why..."

5. 5. Travich 04:09 PM 11/9/10

I don't think that the mechanics of the launching system was the point of this article, since the exact method of launch doesn't matter to the ball once it's been launched. The entire purpose of this article was to explain the initial conditions required of a launched ball to achieve maximum distance. You can have the simplest or most complex launching system you want with the most or least efficiency as long as, at the point of launch, you've generated the greatest initial velocity you can at a 45 degree angle upward from horizontal. As for the other conditions, such as humidity and temperature, I think we can safely say that they are not controllable by the kicker and are not likely to change in any appreciable way within the time allotted for a kick, so they are essentially constant for this experiment. On the other hand, the football is an irregular shape and has three degrees of freedom within any given flight path (pitch, roll, and yaw) that can be controlled to some extent by the kicker. So the question comes down to, How should the ball be oriented initially and what amount of each type of spin will best minimize drag on the ball over its entire flight path? That's computationally more difficult, and may have an effect on what the initial launch angle should be, since drag always acts in the opposite direction to flight path, the direction of which changes in relation to gravity over the entire flight.

6. 6. Lance 06:42 PM 11/9/10

This seems to be a bunch of blither. No disrespect for inquiring minds and I encourage inquiring minds.

The reality is that footballs when kicked travel through a medium called air (or, if you prefer, the atmosphere). The balls are subject to a force called gravity. These two factors plus the shape of the ball and the strength and direction of a kick (not to mention the effect of winds, spin, tumbling, etc.) will determine where the ball lands. This cannot be explained in a few paragraphs.

As any mechanical or aeronautical engineer will tell you, the math is not as simple as graduates of Math 101 or Physics 101 might have you believe. The actual math is manageable but very complex. Parabolas? Ellipses? How simplistic! This is why there is a profession called engineering.

It appears that some commentators have seen fit to spend a lot of time alluding to these issues. But, to what avail?

7. 7. krabcat 08:42 PM 11/9/10

i appreciate that SI puts articles like this on the website. yes it is very simplified, but not everyone will understand the complex equations that would be needed to explain it with full accuracy. as an example, gravity is not really a constant, but with the height that the ball would reach, it would experience approximately 0.00003m/s^2 change in acceleration due to gravity between the ground and the apex, in other words 0.00003%

getting back to the point, in case you skipped the last part, it is close enough to satisfy most people.

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