On an inclined plane, positioned at an angle $\alpha$ to the horizontal, a point mass can slide in the presence of dry friction. The coefficient of friction is denoted by $\mu$. The plane is fixed, and the point moves in such a way that it never loses contact with it.

At any moment in time, the velocity vector $\vec{v}$ of the point mass is decomposed into two perpendicular components: $v_x\vec{e}_x$ — the horizontal velocity, and $v_y\vec{e}_y$ — the velocity of steepest descent. Here $\vec{e}_x,~\vec{e}_y$ are unit vectors in the horizontal direction and in the direction of steepest descent, respectively. That is \[\vec{v} = v_x\vec{e}_x + v_y\vec{e}_y,~~~v = \sqrt{v_x^2 + v_y^2}.\]

It turns out that for the described motion, there exists a nontrivial quantity $I$ that is conserved over time, composed of the velocity components. Up to unknown positive constants $A,~a,~B,~b$, this quantity has the form
\[I= v_x^{a} \left(1 + A\left(\frac{v_y}{v_x}\right)^{b} + B \frac{v_y}{v_x}\sqrt{1 + \left(\frac{v_y}{v_x}\right)^2}\right) = \mathrm{const}.\]

Moreover, the quantities $A,~B,~b$ are the same for any values of $\mu,~\alpha$, while the quantity $a$ is a function of these two variables: \[a = a(\mu,~\alpha).\]

A1  ^1.00 Explain why $a$ is actually a function of a single variable $\gamma = \frac{\mu}{\mathrm{tg}(\alpha)}.$

We will seek the function $a(\gamma)$ in the form
\[a(\gamma) = C\cdot\gamma^{\beta}\]for some constant real quantities $C,~\beta$.

A2  ^2.00 Based on qualitative explanations and theoretical calculations, determine the unique value of $\gamma$ at which one of the terminal velocities $v_x$ or $v_y$ in the steady‑state regime is constant, non‑zero, and finite.

A3  ^3.00 Based on this program, the processing of the obtained results and/or theoretical calculations, reconstruct the functional dependence $a(\gamma)$.

A4  ^4.00 Based on this program, the processing of the obtained results and/or theoretical calculations, reconstruct the values of the constants $A,~B,~b$.

Manual for working with the program.

Enter the value of the friction coefficient, then the angle of inclination of the plane in degrees. Then $v_x = u$, $v_y = w$. Enter the values of the initial conditions $u_0, w_0$, and then you will get a table of values for $u, w$.