$1.3.7.$ A ball is launched along a smooth inclined plane at a speed of $v$. How much horizontal distance will it travel before it rolls off the plane? The plane is inclined to the horizon at an angle of $45^\circ$ . The initial velocity of the ball forms an angle of $45^\circ$ with the horizontal edge of the plane.
On an inclined plane, the body is acted upon by the component of acceleration due to gravity: $$ a = g^* = g \cdot \cos{45^{\circ}} $$ From the formula for flight range obtained in 1.3.6 it follows: $$ L=\frac{v^2}{g^*}\operatorname{sin}90^{\circ} $$ We substitute $g^*$: $$ \fbox{$L=\frac{v^2 \sqrt{2}}{g}$} $$
$$L= \sqrt {2} v^{2} / g$$