The position of a particle *P* on a line *l* is given by 

*s*(*t*) = *t*2 − 6*t*,

 where 

*s*(*t*)

 is in feet and *t* is in seconds

To analyze the position of the particle \( P \) on a line given by \( s(t) = t^2 - 6t \), we can determine various aspects such as velocity, acceleration, and the time at which the particle changes direction. Let's go through these steps:

1. **Position function:**
   \[
   s(t) = t^2 - 6t
   \]

2. **Velocity function:**
   The velocity \( v(t) \) is the derivative of the position function \( s(t) \) with respect to time \( t \):
   \[
   v(t) = \frac{ds(t)}{dt} = \frac{d}{dt}(t^2 - 6t) = 2t - 6
   \]

3. **Acceleration function:**
   The acceleration \( a(t) \) is the derivative of the velocity function \( v(t) \):
   \[
   a(t) = \frac{dv(t)}{dt} = \frac{d}{dt}(2t - 6) = 2
   \]

4. **Time when the particle changes direction:**
   The particle changes direction when its velocity is zero. Setting \( v(t) = 0 \):
   \[
   2t - 6 = 0 \implies 2t = 6 \implies t = 3 \text{ seconds}
   \]

5. **Position at \( t = 3 \):**
   To find the position of the particle at the time it changes direction:
   \[
   s(3) = 3^2 - 6 \cdot 3 = 9 - 18 = -9 \text{ feet}
   \]

6. **Summary of key points:**
   - The velocity function is \( v(t) = 2t - 6 \).
   - The acceleration function is \( a(t) = 2 \), indicating constant acceleration.
   - The particle changes direction at \( t = 3 \) seconds.
   - At \( t = 3 \) seconds, the position of the particle is \( s(3) = -9 \) feet.

Would you like more details or have any questions?

Here are some related questions you might want to ask:

1. How do you find the maximum or minimum position of the particle?
2. What is the significance of the acceleration being constant?
3. Can you explain how to determine if a particle is speeding up or slowing down?
4. How do you find the total distance traveled by the particle in a given time interval?
5. What is the graphical representation of the position, velocity, and acceleration functions?

**Tip:** When analyzing motion, always start by finding the velocity and acceleration functions through differentiation, and use these to determine key events like changes in direction and speed.

Explore the position, velocity, and acceleration of a particle moving along a line given by s(t) = t^2 - 6t. Understand the velocity and acceleration functions, when the particle changes direction, and its position at key moments. Suitable for college-level students and enthusiasts of calculus and kinematics.

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