If the position of a particle at time t is given by s(t) = e 1
3 t3− 3
2 t2+2t+4 for t ≥ 0, find the times for
which the particle is at rest or explain why there are none.

To determine when the particle is at rest, we need to find when its velocity is zero. The velocity of the particle is given by the derivative of its position function \( s(t) \) with respect to time \( t \).

Given:
\[
s(t) = \frac{1}{3}t^3 - \frac{3}{2}t^2 + 2t + 4
\]

1. **Find the derivative** of \( s(t) \) to get the velocity function \( v(t) \):
   \[
   v(t) = \frac{d}{dt} \left( \frac{1}{3}t^3 - \frac{3}{2}t^2 + 2t + 4 \right)
   \]
   Differentiating term by term:
   \[
   v(t) = t^2 - 3t + 2
   \]

2. **Set the velocity function equal to zero** to find the values of \( t \) when the particle is at rest:
   \[
   t^2 - 3t + 2 = 0
   \]
   This is a quadratic equation, which we can factor:
   \[
   (t - 1)(t - 2) = 0
   \]
   Therefore:
   \[
   t = 1 \quad \text{and} \quad t = 2
   \]

3. **Conclusion**: The particle is at rest at \( t = 1 \) and \( t = 2 \).

So, the particle is at rest at \( t = 1 \) and \( t = 2 \) for \( t \geq 0 \).

Would you like further details on this process, or do you have any additional questions?

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### Related Questions:
1. How do you determine if the particle is moving forward or backward at \( t = 1 \) and \( t = 2 \)?
2. What is the acceleration of the particle at the times it is at rest?
3. How would you find the total distance traveled by the particle over a given interval?
4. How do you interpret the behavior of the particle if \( v(t) \) changes sign?
5. Can you find the maximum or minimum position of the particle on a given interval?

### Tip:
To check the behavior of a particle over time, always start by finding its velocity and acceleration, as these indicate motion and changes in speed or direction.

If the position of a particle at time t is given by s(t) = (1/3)t^3 - (3/2)t^2 + 2t + 4 for t ≥ 0, find the times for which the particle is at rest or explain why there are none.

Explore the problem of determining when a particle is at rest based on its position function s(t) = (1/3)t^3 - (3/2)t^2 + 2t + 4. This solution includes finding the velocity function and setting it to zero to discover that the particle is at rest at t = 1 and t = 2.

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