What is the maximum distance that a ball can be thrown in a sports hall with a height of 8 m

What is the maximum distance that a ball can be thrown in a sports hall with a height of 8 m if the initial velocity of the ball is 20 m / s?

h = 8 m.

g = 10 m / s2.

V0 = 20 m / s.

L -?

The maximum distance of the ball throw will be when the ball, during its movement, does not reach the ceiling, that is, it does not rise above h = 8 m.

The movement of the ball can be divided into two types: horizontally, it moves uniformly with a speed V0х, vertical movement is uniformly accelerated, with an acceleration of gravity g and an initial speed V0у.

V0х = V0 * cosα.

V0у = V0 * sinα.

The flight range L is expressed by the formula: L = V0 * cosα * t, where V0 is the ball throwing speed, t is the ball flight time.

g = V0 * sinα / t1, where t1 is the time for lifting the ball to the maximum height h.

t1 = V0 * sinα / g.

t = 2 * t1 = 2 * V0 * sinα / g.

L = V0 * cosα * 2 * V0 * sinα / g = V0 ^ 2 * sin2α / g.

h = V0 ^ 2 * sin2α / 2 * g.

sin2α = 2 * g * h / V0 ^ 2.

sinα = √ (2 * g * h) / V0.

∠α = 39 °.

V0 ^ 2 / g = 2 * h / sin2α.

L = 2 * h * sin2α / sin2α.

L = 2 * 8 m * sin78 ° / sin239 ° = 39 m.

L = 2 * 8 m * sin90 ° / sin245 ° = 39 m.

Answer: the maximum range of the ball is L = 39 m.