1. Understand the Problem

We have a projectile motion problem. The basketball is launched at an angle, and we need to find the initial velocity that will make it reach the hoop.

2. Define Variables

* Initial height (y₀): 2.0 m

* Horizontal distance (x): 10 m

* Final height (y): 3.05 m

* Launch angle (θ): 40°

* Initial velocity (v₀): This is what we need to find.

* Acceleration due to gravity (g): -9.8 m/s² (negative since it acts downwards)

3. Set Up Equations

We'll use the following equations of motion for projectile motion:

* Horizontal motion: x = v_0x * t

* v_0x = v₀ * cos(θ)

* Vertical motion: y = y₀ + v_0y * t + (1/2) * g * t²

* v_0y = v₀ * sin(θ)

4. Solve for Time (t)

* Find the time of flight (t) using the horizontal motion equation:

* t = x / v_0x = x / (v₀ * cos(θ))

5. Substitute Time into the Vertical Motion Equation

* Substitute the expression for 't' from step 4 into the vertical motion equation:

* y = y₀ + v₀ * sin(θ) * (x / (v₀ * cos(θ))) + (1/2) * g * (x / (v₀ * cos(θ)))²

* Simplify the equation:

* y = y₀ + x * tan(θ) + (1/2) * g * (x² / (v₀² * cos²(θ)))

6. Solve for Initial Velocity (v₀)

* Rearrange the equation to solve for v₀:

* v₀² = (g * x² / (2 * (y - y₀ - x * tan(θ)) * cos²(θ)))

* v₀ = √(g * x² / (2 * (y - y₀ - x * tan(θ)) * cos²(θ)))

7. Plug in the Values and Calculate

* Substitute the known values into the equation:

* v₀ = √(9.8 m/s² * (10 m)² / (2 * (3.05 m - 2.0 m - 10 m * tan(40°)) * cos²(40°)))

* Calculate the initial velocity:

* v₀ ≈ 11.6 m/s

Therefore, the basketball player needs to throw the ball with an initial velocity of approximately 11.6 m/s to reach the hoop.