This question asks about when the speed of a particle moving along the y-axis is increasing. The position of the particle is given as $y(t) = t^3 - 6t^2 + 9t$. To determine when the speed is increasing, we follow these steps:

1. Determine Velocity and Acceleration

The speed of the particle is increasing when the velocity and acceleration have the same sign.

• Velocity ($v(t)$) is the first derivative of position: $v(t) = \frac{dy}{dt} = 3t^2 - 12t + 9$

• Acceleration ($a(t)$) is the derivative of velocity: $a(t) = \frac{dv}{dt} = 6t - 12$

2. Find Critical Points

• Set $v(t) = 0$ to find critical points for velocity: $3t^2 - 12t + 9 = 0 \implies t^2 - 4t + 3 = 0 \implies (t-3)(t-1) = 0$ So, $t = 1$ and $t = 3$.

• Set $a(t) = 0$ to find critical points for acceleration: $6t - 12 = 0 \implies t = 2$

3. Analyze Signs of Velocity and Acceleration

To determine when the speed is increasing, evaluate the signs of $v(t)$ and $a(t)$ over the intervals defined by $t = 1$, $t = 2$, and $t = 3$:

Intervals: $0 < t < 1$, $1 < t < 2$, $2 < t < 3$, $3 < t < 4$

• For $v(t)$:

  • Factor $v(t) = 3(t-1)(t-3)$:
    • $0 < t < 1$: $v(t) > 0$
    • $1 < t < 3$: $v(t) < 0$
    • $t > 3$: $v(t) > 0$

• For $a(t)$:

  • $a(t) = 6(t-2)$:
    • $0 < t < 2$: $a(t) < 0$
    • $t > 2$: $a(t) > 0$

Speed Increasing:

The speed increases when $v(t)$ and $a(t)$ have the same sign:

• $1 < t < 2$: Both $v(t) < 0$ and $a(t) < 0$.
• $3 < t < 4$: Both $v(t) > 0$ and $a(t) > 0$.

4. Answer

The correct intervals are $1 < t < 2$ and $3 < t < 4$. Therefore, the answer is:

$\boxed{\text{D: } 1 < t < 2 \text{ and } 3 < t < 4}$

Would you like a step-by-step explanation for any part?

Related Questions:

1. How do you determine the critical points for a given polynomial?
2. Why do the signs of velocity and acceleration determine when speed is increasing?
3. How can you graphically represent $v(t)$ and $a(t)$?
4. Can speed ever decrease if $v(t)$ and $a(t)$ have opposite signs?
5. How does this problem extend to higher dimensions of motion?

Tip:

Always check the relationship between velocity and acceleration when analyzing the change in speed.