$8.3.9.$ [Insert problem description here]
Example Statement:
$1.1.1.$ Determine the coordinate $x(t)$ of a body as a function of time $t$, given that its acceleration is defined as $a(t) = bt$, where $b$ is a constant.
[Your solution should be placed here]
Example Solution:
The acceleration of the body defined by
$$a(t) = bt$$
We know that acceleration is the time derivative of velocity:
$$a(t) = \frac{d v(t)}{d t}$$
To find the velocity $v(t)$, we integrate $a(t)$ with respect to time:
$$v(t) = \int a(t) \, dt = \int b t \, dt$$
If the initial velocity is $v(0) = 0$, then the velocity becomes:
$$v(t) = \frac{b t^2}{2}$$
Likewise, integrate $v(t)$ with respect to time:
$$x(t)= \int v(t) \, dt = \frac{b}{2} \int t^2 \, dt$$
From where the coordinate from time, considering the initial conditions:
$$\boxed{x(t)=\frac{bt^3}{6}}$$
[Insert a concise answer or boxed result, like this:]
Example Answer:
$$ x(t)=\frac{bt^3}{6} $$