To find the instantaneous velocity of an object moving with rectilinear motion, we'll use the definition of instantaneous velocity, which is the derivative of the position function $s(t)$ with respect to time $t$.

Step 1: Position function

The given position function is: $s(t) = 3t^2 - 4t$

Step 2: Instantaneous velocity as the derivative

The instantaneous velocity $v(t)$ is the derivative of $s(t)$ with respect to $t$: $v(t) = \frac{ds(t)}{dt}$

Step 3: Differentiate the position function

Differentiate $s(t) = 3t^2 - 4t$ with respect to $t$: $v(t) = \frac{d}{dt}(3t^2 - 4t)$ Using the power rule: $v(t) = 6t - 4$

Step 4: Calculate the instantaneous velocity at $t = 6$

Now, substitute $t = 6$ into the velocity function to find the instantaneous velocity: $v(6) = 6(6) - 4$ $v(6) = 36 - 4$ $v(6) = 32 \text{ ft/s}$

Final Answer:

The expression for the instantaneous velocity is $v(t) = 6t - 4$, and the instantaneous velocity at $t = 6$ seconds is $32$ ft/s.

Would you like more details, or do you have any questions?

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Here are some related questions:

1. How would you find the instantaneous velocity if the position function was $s(t) = t^3 - 4t^2 + 7t$?
2. What is the physical meaning of the derivative in the context of rectilinear motion?
3. How do you find the instantaneous velocity at a specific time for a position function given in terms of a trigonometric function, such as $s(t) = \sin(t)$?
4. What is the significance of the sign of the instantaneous velocity?
5. How would you find the acceleration of the object from the position function $s(t) = 3t^2 - 4t$?
6. How do you determine the time at which the object is momentarily at rest?
7. What is the difference between average velocity and instantaneous velocity?
8. How does the concept of limits relate to finding instantaneous velocity?

Tip: When calculating instantaneous velocity, make sure the position function is fully simplified before taking the derivative to avoid errors.