The longest pass in NFL history is approximately 83.7 m. Ignoring the effects of air resistance, calculate what must have been the initial speed of the football.
To calculate the initial speed of the football, we can use the equation of motion for horizontal projectile motion:
Range = (v^2 * sin(2θ)) / g
Where:
- Range is the horizontal distance travelled by the football (83.7 m in this case).
- v is the initial speed of the football.
- θ is the launch angle (which we'll assume to be 45 degrees for maximum range).
- g is the acceleration due to gravity (approximately 9.8 m/s^2).
Rearranging the equation to solve for v:
v^2 = (Range * g) / sin(2θ)
Taking the square root of both sides:
v = √((Range * g) / sin(2θ))
Plugging in the given values:
v = √((83.7 * 9.8) / sin(2 * 45))
Calculating further:
v = √((83.7 * 9.8) / sin(90))
Since sin(90) is equal to 1:
v = √(83.7 * 9.8)
v ≈ √819.66
v ≈ 28.64 m/s
Therefore, the approximate initial speed of the football would be 28.64 m/s.
To calculate the initial speed of the football, we can use the following formula:
v^2 = u^2 + 2as
Where:
- v is the final velocity of the football (which we can assume to be zero as the football reaches its maximum height)
- u is the initial velocity or speed of the football (what we need to find)
- a is the acceleration due to gravity (approximately 9.8 m/s^2)
- s is the distance traveled (83.7 m)
Given that the final velocity is 0, we have:
0 = u^2 + 2(9.8)(83.7)
Simplifying the equation:
0 = u^2 + 1631.16
Rearranging the equation:
u^2 = -1631.16
To solve for u, we can take the square root of both sides:
u = √(-1631.16)
However, we encounter a problem here because the square root of a negative number is not a real number. This means that it is not possible to determine the initial speed of the football without considering the effects of air resistance.