So if the initial velocity is +5, then the final velocity has to be -5. However, we should easily see that the projectile was at first going up, but then it finishes by going down, thus we have to write the y component of the final velocity with the opposite sign of the y component of the initial velocity. And since the starting and ending points have the same elevation, we can then assume that the projectile has equal speed at those two points. We assume this to be true since we are also assuming that there is no air resistance. So we choose the final velocity to be just before it hits the ground.Īnd what is the final velocity before it hits the ground? Well, the projectile does not lose any energy while from the time right after it is launched to the time just before it lands. Fortunately, this problem can be solved just with the motion of the projectile before it hits the ground, so we don't need to concern ourselves with anything after that. Then only after it hits the ground will it have zero velocity, but hitting the ground will introduce another force to this system, and we would need to use more equations to describe its motion. Just before it hits the ground, the projectile has some downward speed. So we should only apply them to the motion of the projectile right after it is thrown and right before it hits the ground. This means that the only force acting on it is the force of gravity. Not on purpose, anyway.The equations that we are using to solve this problem only apply when the projectile is in free fall. 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. 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.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |