$1.4.5.$ a. Due to air resistance, raindrops fall at a constant velocity $v$ perpendicular to the ground surface. How is it necessary to position a cylindrical bucket on a platform moving at a speed of $u$ so that drops do not fall on its walls?
b. At a wind speed of $10$ $\frac{m}{s}$, raindrops fall at an angle of $30^\circ$ to the vertical. At what wind speed will the drops fall at an angle of $45^\circ$?
a) To prevent raindrops from hitting the walls of the bucket, their relative velocity must be parallel to the walls of the bucket.
In the inertial reference frame associated with raindrops, the angle $\alpha$ between the relative velocity and the horizontal is defined as
$$\tan \varphi = u/v \; (1)$$
The walls of the bucket must be at the same angle.
b) Considering the relationship of the angle with the vertical and horizontal $\beta = 90^{\circ} - \alpha$. We substitute numerical data into $(1)$ and solve the system:
$$\left\{\begin{matrix} 10 \cdot \tan 60^{\circ} = u\\ v \cdot \tan 45^{\circ} = u \end{matrix}\right.$$
$a.$ The bucket should be tilted in the direction of the platform's movement at an angle $\varphi$ to the vertical: $\tan \varphi = u/v.$
$б.$ $u = 10\sqrt{3} \, m/s$