![range of projectile formula range of projectile formula](https://i.ytimg.com/vi/YmU_LS16GTc/maxresdefault.jpg)
The time for the ball to reach maximum level is S = flight distance (m, ft) Example - Throwing a BallĪ ball is thrown with initial velocity 25 m/s in angle 30 degrees to the horizontal plane. The horizontal distance of the flight can be expressed as The maximum elevation - h - of the flight can be calculated as The total flight time can be expressed as Θ = the initial angle of the velocity vector to the horizontal plane (degrees)Ī g = acceleration of gravity (9.81 m/s 2, 32.174 ft/s 2) V i = initial velocity of the projectile (m/s, ft/s) Not on purpose, anyway.The time for a projectile - a bullet, a ball or a stone or something similar - thrown out with an angle Θ to the horizontal plane - to reach the maximum height can be calculated as 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. 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. 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. 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. (Anyone looking to loft a projectile as high as possible would simply launch it as fast as possible, and gravity is constant.) Once again, we can ignore v and g, for the same reasons as above. A projectile, in other words, travels the farthest when it is launched at an angle of 45 degrees.īut 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. 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.
![range of projectile formula range of projectile formula](https://i.ytimg.com/vi/bqYtNrhdDAY/hqdefault.jpg)
![range of projectile formula range of projectile formula](https://i.ytimg.com/vi/wvc6UcG4nS0/maxresdefault.jpg)
![range of projectile formula range of projectile formula](https://i.ytimg.com/vi/UkLZKxxTFaA/hqdefault.jpg)
You can see from the equation above that the distance traveled by the ball will be greatest when sin(2θ) is greatest. The only choice he has to make to maximize distance, then, is the angle at which he 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. First, because the force of gravity is constant, g will be the same no matter how a punter kicks the ball. That may look like a complicated equation, but a couple of the variables can be ignored. 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. Parabolas have been studied for millennia, and their properties are well understood. (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.) 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. Earth's gravitational pull makes long-range passing a challenge and pulls down even the hardest-struck punts and placekicks.īecause gravity is a constant, experienced quarterbacks and kickers can account for its effects to move the ball downfield as efficiently as possible. In any football game both teams square off against each other and against a shared opponent as well-gravity. 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.