$1.1.1.$ 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.
### Solution
[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:
$1.1.1.$ 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.
$1.1.1.$ 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.
### Solution
### Solution
[Your solution should be placed here]
[Your solution should be placed here]
__Example Solution__:
__Example Solution__:
The acceleration of the body defined by
The acceleration of the body defined by
$$a(t) = bt$$
$$a(t) = bt$$
We know that acceleration is the time derivative of velocity:
We know that acceleration is the time derivative of velocity:
$$a(t) = \frac{d v(t)}{d t}$$
$$a(t) = \frac{d v(t)}{d t}$$
To find the velocity $v(t)$, we integrate $a(t)$ with respect to time:
To find the velocity $v(t)$, we integrate $a(t)$ with respect to time:
$$v(t) = \int a(t) \, dt = \int b t \, dt$$
$$v(t) = \int a(t) \, dt = \int b t \, dt$$
If the initial velocity is $v(0) = 0$, then the velocity becomes:
If the initial velocity is $v(0) = 0$, then the velocity becomes:
