3. From a point on the Earth's surface, a large number of small balls are launched in all directions at the same speed of 10 m/s. Among all the balls that land at a distance from the starting point not closer than $96 \%$ of the distance at which the farthest flying ball lands, find the one that will spend the most time in flight.

What is this time? Express the answer in seconds and round to one decimal place. The acceleration due to gravity is $10 \mathrm{~m} / \mathrm{c}^{2}$.
Solution. The range of a body thrown with an initial velocity $V_{0}$ at an angle $\alpha$ to the horizontal is $l=\frac{V_{0}^{2} \sin 2 \alpha}{g}$, and the flight time is $\tau=\frac{2 V_{0} \sin \alpha}{g}$.

According to the condition $\frac{V_{0}^{2} \sin 2 \alpha}{g} \geqslant \frac{96}{100} \frac{V_{0}^{2} \sin \left(2 \cdot 45^{\circ}\right)}{g}$, i.e., $\sin 2 \alpha \geqslant \frac{96}{100}$. Among these balls, we need to choose the one with the maximum $\tau=\frac{2 V_{0} \sin \alpha}{g}$, i.e., the one with the largest $\sin \alpha$. Since $\arcsin \frac{24}{25} \leqslant 2 \alpha \leqslant \pi-\arcsin \frac{24}{25}$, then $\frac{1}{2} \arcsin \frac{24}{25} \leqslant \alpha \leqslant \frac{\pi}{2}-\frac{1}{2} \arcsin \frac{24}{25}$. Therefore, the maximum flight time is $\frac{2 V_{0}}{g} \sin \left(\frac{\pi}{2}-\frac{1}{2} \arcsin \frac{24}{25}\right)=\frac{2 V_{0}}{g} \cos \left(\frac{1}{2} \arcsin \frac{24}{25}\right)$ $=\frac{2 V_{0}}{g} \sqrt{\frac{1+\cos \left(\arcsin \frac{24}{25}\right)}{2}}=\frac{2 V_{0}}{g} \sqrt{\frac{1+\frac{7}{25}}{2}}=\frac{2 V_{0}}{g} \sqrt{\frac{16}{25}}=1.6 \text{s}$.
Answer: 1.6 seconds.

Grading criteria: 20 points - correct (not necessarily the same as above) solution and correct answer; $\mathbf{1 5}$ points - mostly correct solution, but with defects (for example, the choice of $\alpha$ is not justified); 10 points - solution is mostly correct, but contains computational errors; $\mathbf{5}$ points - correct formulas for range and flight time are written, but there is no substantial progress.