To solve the problem, we can use the kinematic equation given:

$v_f^2 = v_i^2 + 2ad$

Where:

• $v_f$ is the final velocity, which will be 0 m/s at the maximum height (since the ball stops rising).
• $v_i$ is the initial velocity, 30 m/s (given).
• $a$ is the acceleration, which is the acceleration due to gravity $-9.8 \, \text{m/s}^2$ (negative because it opposes the upward motion).
• $d$ is the displacement, which is the maximum height reached by the ball (what we need to find).

Let's solve for $d$:

$0^2 = (30)^2 + 2(-9.8)d$

This simplifies to:

$0 = 900 - 19.6d$

Now, solve for $d$:

$19.6d = 900$

$d = \frac{900}{19.6} \approx 45.9 \, \text{m}$

So, the maximum height is approximately 46 m, which corresponds to option B.

Would you like more details or have any further questions?

Here are five related questions for further exploration:

1. What would happen to the height if the initial velocity were doubled?
2. How long would it take for the ball to reach its maximum height?
3. How would the result change if gravity were different, say on another planet?
4. Can you calculate the total time the ball is in the air, considering both the upward and downward motion?
5. What if the ball were projected at an angle instead of vertically—how would that affect the height?

Tip: Always pay attention to the signs of the acceleration and velocity, especially when dealing with upward and downward motions.