The amplitude of small oscillations of a mathematical pendulum standing on a cart is x_0, and the amplitude of oscillations of the cart is y_0. The length of the pendulum thread is l.Determine the maximum speed of the pendulum and cart. Neglect friction.

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    <h3> Statement </h3>
    <p>
        $3.1.15.$ The amplitude of small oscillations of a mathematical pendulum standing on a cart is $x_0$, and the amplitude of oscillations of the cart is $y_0$. The length of the pendulum thread is $l$.

Determine the maximum speed of the pendulum and cart. Neglect friction.

    <h3>Solution</h3>
    <p>
        Law of conservation of energy

$$\frac{mg(x_0+y_0)^2}{2l}=\frac{m\upsilon^2}{2}+\frac{MU^2}{2}$$
Law of conservation of momentum
$$m\upsilon =MU$$
Because the center of mass does not change position
$$mx_0=My_0$$
Let's solve the upper equation for $\upsilon$:


$$\upsilon =\sqrt{\frac{Mg}{(M+m)l}(x_0+y_0)^2}=\sqrt{\frac{x_0g}{(x_0+y_0)l}(x_0+y_0)^2}$$
Maximum speed of the pendulum
$$\boxed{v=x_0\sqrt{\frac{g}{l}(1+\frac{y_0}{x_0})}}$$
Let's solve the upper equation for $U$:


$$U=\sqrt{\frac{mg}{(M+m)l}(x_0+y_0)^2}=\sqrt{\frac{y_0g}{(x_0+y_0)l}(x_0+y_0)^2}$$
Maximum speed of the trolley
$$\boxed{U=y_0\sqrt{\frac{g}{l}(1+\frac{x_0}{y_0})}}$$

    </p>

    <h4>Answer</h4>
    <p>
        $$\upsilon =x_0\sqrt{\frac{g}{l}(1+\frac{y_0}{x_0}});\quad U=y_0\sqrt{\frac{g}{l}(1+\frac{x_0}{y_0})}$$
    </p>
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