1. A boy standing on a horizontal ground surface at a distance of $L=5$ m from a vertical wall of a house kicked a ball lying in front of him on the ground. The ball flew at an angle of $\alpha=45^{\circ}$ to the horizon and, after an elastic collision with the wall, fell back to the same place where it initially lay.
1) Find the time of flight of the ball.
2) At what height above the ground did the ball bounce off the wall?

Assume the acceleration due to gravity to be $10 \mathrm{~m} / \mathrm{c}^{2}$.
1. $V_{0} \sin \alpha-g \tau=0 \Rightarrow V_{0}=\frac{g \tau}{\sin \alpha}$

$$
L=V_{0} \cos \alpha \cdot \tau=\frac{g \tau}{\sin \alpha} \cdot \cos \alpha \cdot \tau=\frac{g \tau^{2}}{\operatorname{tg} \alpha} \Rightarrow \tau=\sqrt{\frac{\operatorname{Ltg} \alpha}{g}}
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_556368b0f812d0b7a511g-3.jpg?height=357&width=511&top_left_y=364&top_left_x=1389)

$T=2 \tau=2 \sqrt{\frac{\operatorname{Ltg} \alpha}{g}}=2 \sqrt{\frac{5 \cdot 1}{10}}=\sqrt{2} \approx 1.4 \text{s}$

$H=V_{0} \sin \alpha \cdot \tau-\frac{g \tau^{2}}{2}=g \tau^{2}-\frac{g \tau^{2}}{2}=\frac{g \tau^{2}}{2}=\frac{g}{2} \cdot \frac{\operatorname{Ltg} \alpha}{g}=\frac{L}{2} \operatorname{tg} \alpha=\frac{5}{2} \cdot 1=2.5 \text{m}$