To solve this problem, we need to follow these steps:

Given:

The position of the particle as a function of time is: $x(t) = (17.7 \ \text{m/s})t - \left(\frac{1.8 \ \text{m}}{3}\right) t^3$

We need to find the instantaneous velocity at two specific times and the average velocity between these two times.

Part (a) - Instantaneous velocity at $t = 2.5 \ \text{s}$

The instantaneous velocity is the derivative of the position function with respect to time: $v(t) = \frac{dx(t)}{dt}$

Given $x(t) = 17.7t - 0.6t^3$, differentiate with respect to $t$: $v(t) = 17.7 - 1.8t^2$

Now, plug in $t = 2.5 \ \text{s}$ to find the instantaneous velocity at that time: $v(2.5) = 17.7 - 1.8(2.5)^2$

Part (b) - Instantaneous velocity at $t = 3.9 \ \text{s}$

Similarly, plug in $t = 3.9 \ \text{s}$ into the velocity function: $v(3.9) = 17.7 - 1.8(3.9)^2$

Part (c) - Average velocity between $t = 2.5 \ \text{s}$ and $t = 3.9 \ \text{s}$

The average velocity is given by the change in position divided by the change in time: $v_{\text{avg}} = \frac{x(3.9) - x(2.5)}{3.9 - 2.5}$

First, calculate the positions at $t = 3.9 \ \text{s}$ and $t = 2.5 \ \text{s}$ using the position function $x(t)$.

Let’s calculate all the values step by step.### Part (a) - Instantaneous velocity at $t = 2.5 \ \text{s}$: The instantaneous velocity at $t = 2.5 \ \text{s}$ is $6.45 \ \text{m/s}$.

Part (b) - Instantaneous velocity at $t = 3.9 \ \text{s}$:

The instantaneous velocity at $t = 3.9 \ \text{s}$ is $-9.68 \ \text{m/s}$.

Part (c) - Average velocity between $t = 2.5 \ \text{s}$ and $t = 3.9 \ \text{s}$:

The average velocity between $t = 2.5 \ \text{s}$ and $t = 3.9 \ \text{s}$ is $-1.03 \ \text{m/s}$.

If you have any questions or need further details, feel free to ask!

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Here are 5 related questions:

1. What is the acceleration of the particle at $t = 2.5 \ \text{s}$?
2. How does the velocity change between $t = 2.5 \ \text{s}$ and $t = 3.9 \ \text{s}$?
3. What is the significance of the negative velocity at $t = 3.9 \ \text{s}$?
4. How would the particle's position change if the cubic term had a different coefficient?
5. Can you find the time when the particle comes to rest?

Tip: When working with motion problems, always pay attention to the sign of the velocity, as it indicates the direction of motion.