Calculating Initial Velocity & Angle of Football Kicked on Horizontal Plane

[brunch43]

ok, so i just had my exam 10 minutes ago and i can't think of anything else except this seemly easy problem that i couldn't get.

a football is kicked on the horizontal plane (ie. y_0 = 0 ) at some angle alpha, it covers a horizontal displacement of 100 ft 2.5 seconds later, find the initial velocity and the angle. i feel so f-ing stupid, but i will greatly appreciate any help given

 

[Office_Shredder]

You know the horizontal velocity. Since the vertical velocity is independent, try figuring out high the football can go if it's to land in 2.5 seconds

 

[HallsofIvy]

Assume the initial speed is v. The initial horizontal speed is $v cos(\alpha)$, the initial vertical speed is $v sin(\alpha)$. The (constant) vertical acceleration is -g and the there is 0 horizontal velocity so the horizontal speed is the constant $vcos(\alpha)$ and the vertical speed is $-gt+ vsin(\alpha)$. Integrating those, the horizontal position is $vcos(\alpha)t$ and the vertical position is $-(g/2)t^2+ vsin(\alpha)t$. Knowing that the horizontal distance covered in 2.5 sec. is 100 feet, gives $2.5v cos(\alpha)= 100$, You also know that the ball went up and back down to 0 in that time: the vertical equation gives $-(g/2)(2.5)^2+ 2.5 vsin(\alpha)= 0$. That gives you two equations to solve for v and $\alpha$