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.