Lanzamos desde el suelo una pelota con un ángulo de 45° y queremos colarla en una cesta que está a 7 m de distancia horizontal y a 3.5 m de altura. Calcular con qué velocidad hay que lanzarla.

 v₀ = 11.72 m / s

Explanation:

This is a missile launch exercise. Let's write the relations for the final position

        x = v₀ₓ t

        y = t - ½ g t²

Let's use trigonometry to find the initial velocities

      sin 45 = v_{oy} / v₀

      cos 45 = v₀ₓ / v₀

      v_{oy} = v₀ sin 45

     v₀ˣ = v₀ cos45

we substitute

      x = v₀ cos 45 t

      y = v₀ sin 45 t - ½ g t²

     in this equation system we have two unknowns and two equations for which it can be solved, let's substitute the values

     7 = v₀ 0.707 t

     3.5 = v₀ 0.707 t - ½ 9.8 t²

we clear the time in the first equation and substitute in the second

     3.5 = v₀ 0.707 (7 / v₀ 0.707) - 4.9 (7 / v₀ 0.707) 2

     3.5 = 7 - 480.345 / v₀²

      480.345 / v₀² = 7-3.5

      v₀² = 480.345 / 3.5

      v₀ = √ 137.24

      v₀ = 11.72 m / s

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