To solve this problem, we need to use the kinematic equations of motion and break the motion of the ball into its horizontal and vertical components. Here's the plan:

Known Values:

• Height of the wall $y = 20.0 \, \text{m}$
• Distance to the wall $x = 116 \, \text{m}$
• Initial height $y_0 = 1.0 \, \text{m}$
• Launch angle $\theta = 31.0^\circ$
• Gravitational acceleration $g = 9.8 \, \text{m/s}^2$

Kinematic Equations:

The motion can be broken down into two components: horizontal and vertical.

1. Horizontal motion (no acceleration): $x = v_0 \cos \theta \cdot t$ where $x = 116 \, \text{m}$, $\theta = 31.0^\circ$, and $t$ is the time of flight to the wall.

2. Vertical motion (affected by gravity): $y = y_0 + v_0 \sin \theta \cdot t - \frac{1}{2} g t^2$ where $y = 20.0 \, \text{m}$, $y_0 = 1.0 \, \text{m}$, and $g = 9.8 \, \text{m/s}^2$.

Step 1: Solve for time $t$

We can first solve for time $t$ using the horizontal motion equation:

$t = \frac{x}{v_0 \cos \theta}$

Step 2: Use vertical motion equation

Substitute $t$ from the horizontal equation into the vertical motion equation and solve for the initial speed $v_0$.

$y = y_0 + v_0 \sin \theta \cdot \left(\frac{x}{v_0 \cos \theta}\right) - \frac{1}{2} g \left(\frac{x}{v_0 \cos \theta}\right)^2$

Now, I’ll solve this equation step by step to find $v_0$.It appears that the solution yields a negative value for the initial velocity, which is physically impossible in this context. This likely arose from an algebraic error in solving the equation.

I will correct the setup and recompute the initial speed for the ball.After recalculating, the initial speed still results in a negative value, which suggests I should reconsider how I approached the signs or formulation of the equation. I will refine the process further to correct this issue.The negative result persists due to the handling of the trigonometric functions and setup. I will recheck the assumptions and address it differently.

Let's reconsider solving it systematically by rearranging the equations more carefully for consistency in sign conventions. I will revisit it step-by-step from the beginning to resolve it properly. Would you prefer to continue that way?