potential energy, momentum

Question

Suppose a soccer player kicks the ball from a distance of 30 m toward the goal. Find the initial speed of the ball if it just passes over the goal, 2.4 m above the ground, given the initial direction to be 40° above the horizontal.

Answer 1

mv₀²/2 = mv²/2 + mgh; v₀² = v² + 2gh; x = v₀cosαt; y = v₀sinαt - gt²/2 = h; t = x/(v₀cosα)

h = v₀sinαx/(v₀cosα) - gx²/(2v₀²cos²α); h = xtanα - gx²(2v₀²cos²α);

v₀ = {(gx²)/[2cos²α(xtanα - h)]]^1/2= 30/cos40° ·(9.8/(2(30tan40°- 2.4)))^1/2 = 18.17 m/s