The total mass of the vehicle without the wheels $M$ is equally distributed between the wheels axes. The mass of the each wheel is $m$ and the length of its square side is $2a$.

Part A. Energy of the motion. (2.5 points)

A1 Determine the moment of inertia of the wheel with the respect to its central axis. Express the answer in terms of $a$ and $m$.

A S

The vehicle's movement starts in the tops of the two road bumps with initial horizontal velocity $v_0$.

A2

Determine the vehicle's total kinetic energy $E_{\text{kin}}$ and the angular velocity $\omega_T$ of the wheels in the tops of the road bump. Express the answer in the terms of $a$, $m$, $M$ and $v_0$. 

A S

There are no dissipative forces under consideration, therefore the total mechanical energy of the vehicle is conserved. Moreover, the center of mass of the vehicle does not move vertically, hence the kinetic energy $E_{\text{kin}}$ of the vehicle is conserved. However, because of the wheel and the road contact axis is constantly changing during the motion, the moment of inertia of the wheels with respect to the contact axis and its angular velocity is changing.

A3

Using the energy conservation law, determine the angular velocity $\omega_{V}$ of the wheels in the valley. Express the answers in terms of $a$, $m$, $M$ and $v_0$.

A S

Part B. The shape of the road. (4.5 points)

The motion of the wheel across the road is illustrated in the figure above. The road is the periodical repetition of identical bumps. $AB$ — the edge of the wheel's square side. $T$ — the contact axis of the road and the wheel. $G$ — the center of the wheel. $A^\prime, B^\prime, T^\prime, G^\prime$ — the same but for the arbitrary moment in time. $x_s$ and $x_d$ — the coordinates of the extreme points of the bump. Now it is time to determine the analytical formula for the shape of the road. 

B1 Write down the equation relating the coordinate of the road bump surface $y(x)$ and its slope angle $\alpha$.

A S

Precise calculations show that the equation for the analytical shape of the road is of the form:

\[ y(x) = k - h\cdot \dfrac{e^{x/a}+e^{-x/a}}{2}, \]

where $k$ and $h$ are the parameters depending only on $a$.

B2 Using the previous form for the $y(x)$, express the slope of the road $\tan \alpha(x)$ in terms of $k$, $h$, $a$, $x$.

A S

B3 Substitute the formulae of the $y(x)$ and $\tan \alpha(x)$ into the equation obtained in B1. Determine the values of the parameters $k$ and $h$. Express the answer in terms of $a$.

A S

B4 Determine the horizontal length of one road bump $d$.

A S

Part C. Any point in time. (3 points)

One who knows the shape of the road $y(x)$ can determine the wheel rotating radius in arbitrary moment in time. Using the radius, one can obtain the moment of inertia of the wheel with the respect to the contact axis. Now using the $y(x)$ formula and the energy conservation law determine:

C1 Angular velocity of the wheel as a function if the road horizontal coordinate $\omega(x)$. Express the answer in terms of $a$, $m$, $M$ and $v_0$.

A S

C2 Horizontal velocity of the vehicle as a function if the road horizontal coordinate $v(x)$. Express the answer in terms of $a$, $m$, $M$ and $v_0$.

A