$1.3.25.$ A small body moves along a circle of radius $r$ at a speed that increases linearly in time according to the law $v = kt$. Find the dependence of the total acceleration of the body on time.
We will decompose the acceleration into two components, tangential $a_\parallel $ and normal $a_\perp$
In the longitudinal direction, the acceleration $a_\parallel $ is associated with the change in speed
$$a_\parallel = k$$
In the transverse direction, the acceleration $a_\perp $ is a consequence of the centrifugal force
$$ a_\perp = \frac{v^2}{r} = \frac{k^2t^2}{r} $$
Full acceleration
$$a = \sqrt{a^2_\perp + a^2_\parallel}$$
Substitute the obtained components
$$ \boxed{a = \sqrt{\frac{k^4t^4}{r^2} + k^2}} $$
$$a = \sqrt{\frac{k^4t^4}{r^2} + k^2}$$