The given problem describes the motion of an object with the position function $s(t) = 1 - 13t + 7t^2$, where $s(t)$ is the distance from the origin at time $t$.

1. Find the velocity $v(t)$ of the object at any time $t$:

To find the velocity, $v(t)$, we need to take the derivative of the position function $s(t)$ with respect to time $t$.

$v(t) = \frac{d}{dt} s(t)$

Given $s(t) = 1 - 13t + 7t^2$, differentiate term by term:

$v(t) = \frac{d}{dt}(1) - \frac{d}{dt}(13t) + \frac{d}{dt}(7t^2)$ $v(t) = 0 - 13 + 14t$ Thus, the velocity function is:

$v(t) = 14t - 13$

2. When is the object at rest?

The object is at rest when its velocity is zero. To find when the object is at rest, set $v(t) = 0$:

$14t - 13 = 0$ $14t = 13$ $t = \frac{13}{14}$

Thus, the object is at rest at $t = \frac{13}{14}$ seconds.

Would you like more details on these steps or have any other questions?

Here are 5 related questions to explore further:

1. What is the acceleration of the object at any time $t$?
2. What is the object's position at $t = 0$ and $t = 1$?
3. How far does the object travel by the time it comes to rest?
4. What is the total displacement between $t = 0$ and $t = 2$?
5. What type of motion (speeding up or slowing down) is the object experiencing before it comes to rest?

Tip: The velocity is the derivative of position, and acceleration is the derivative of velocity, which helps you analyze how an object moves over time.