13.273. The ball is rolling perpendicular to the sideline of a football field. Suppose, moving uniformly decelerated, the ball rolled 4 m in the first second, and 0.75 m less in the next second. A football player, initially 10 m away from the ball, started running in the direction of the ball's movement to catch it. Moving uniformly accelerated, the player ran 3.5 m in the first second, and 0.5 m more in the next second. How long will it take for the player to catch the ball, and will he be able to catch it before the ball goes out of bounds, if the player needs to run 23 m to reach the sideline?
Solution.

Let $t$ be the desired time. In this time, the ball will roll $\frac{2 \cdot 4-0.75(t-1)}{2} \cdot t \mathrm{M}$, and the footballer will run $\frac{2 \cdot 3.5+0.5(t-1)}{2} \cdot t \mathrm{M}$. (The formula for the sum of $n$ terms of an arithmetic progression was used).

According to the condition,

$$
\frac{2 \cdot 3.5+0.5(t-1)}{2} \cdot t - \frac{2 \cdot 4-0.75(t-1)}{2} \cdot t = 10, \text{ from which } t = 5 \mathrm{s}
$$

The footballer will catch up with the ball, having run $\frac{2 \cdot 3.5+0.5 \cdot 4}{2} \cdot 5 = 22.5$ m, 0.5 m before the sideline.

Answer: in 5 s; 0.5 m before the line of the field.