The body starts moving from point A and moves first equidistant for time t_0, then with the same modulo acceleration — equidistant. After what time from the beginning of the movement, the body will return to point A?

Solutions of Savchenko Problems in Physics

Aliaksandr Melnichenka
October 2023

    <h3 id="back-link"><a href="/#1.2">$\leftarrow$Back</a></h3>

    <h3> Statement </h3>
    <p>
        $1.2.20.$ The body starts moving from point $A$ and moves first equidistant for time $t_0$, then with the same modulo acceleration — equidistant. After what time from the beginning of the movement, the body will return to point $A$?
    </p>

    <h3>Solution</h3>
    <p>
        During the time $t_0$ the velocity of the body became equal to $v_0 = at_0$, and the coordinate 

$$x_0 = \frac{at_0^2}{2}$$
It takes time $t_1$ for the body to travel the distance $x_1$:
$$-x_1 = v_0 t - \frac{at^2}{2}$$
Solving a quadratic equation:
$$t_1^2 - 2t_0 t_1-t_0^2=0$$
We choose a positive root:
$$t_1 = (1+\sqrt{2})t_0$$
Round trip time:
$$T = t_0+t_1=(2+\sqrt{2})t_0$$

    </p>

    <h4>Answer</h4>
    <p>
        $$t =(2+\sqrt{2})t_0$$
    </p>


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