1. **Problem Statement:** A baseball is thrown straight upward with an initial velocity of $25$ m/s. We need to find the total time elapsed for the baseball to return to the thrower's hand, assuming zero air resistance. 2. **Relevant Formula:** The motion is under constant acceleration due to gravity, $g = 9.8$ m/s$^2$ downward. The total time to go up and come back down is given by: $$t = \frac{2v_0}{g}$$ where $v_0$ is the initial velocity. 3. **Explanation:** The baseball first rises until its velocity becomes zero at the peak, then falls back down. The time to reach the peak is $\frac{v_0}{g}$, and the total time is twice that. 4. **Calculation:** $$t = \frac{2 \times 25}{9.8} = \frac{50}{9.8} \approx 5.10 \text{ seconds}$$ 5. **Interpretation:** The baseball takes approximately $5.10$ seconds to return to the starting point. **Final answer:** A. 5.10 sec