$1.2.3.$ A body for time $t_0$ moves with constant velocity $v_0$. Then its velocity increases linearly with time so that at time $2 t_0$ it is equal to $2 v_0$. Determine the path traveled by the body for time $t > t_0$.
At time $t_0$, the coordinate was $x_1=v_0t_0$.
On the interval from $t_0$ to $2t_0$, the acceleration is constant and equal to $a=\frac{v_0}{t_0}$.
In the case of equiaxcelerated motion with initial velocity $v_0$ from time $t_0$, the path is found as
$$ x_2 = v_0 (t-t_0)+\frac{a(t-t_0)^2}{2} $$
$$ x = x_1+x_2 $$
$$ x = v_0 t+\frac{a(t-t_0)^2}{2} $$
$$L = v_{0}t + \frac{v_{0} (t − t_{0})^{2}}{2t_{0}}$$