The position of an object moving along a straight line is given by the function $s(t) = 4t^3 - 3$. The average velocity over a time interval $[t_1, t_2]$ is defined as:

$v_{\text{avg}} = \frac{s(t_2) - s(t_1)}{t_2 - t_1}$

Step-by-Step Process:

We will calculate the average velocity over the given time intervals and then estimate the instantaneous velocity at $t = 2$.

1. Average velocity over $[2, 2.1]$:

• $t_1 = 2$, $t_2 = 2.1$
• $s(2) = 4(2)^3 - 3 = 32 - 3 = 29$
• $s(2.1) = 4(2.1)^3 - 3 = 4(9.261) - 3 = 37.044 - 3 = 34.044$
• Average velocity:

$v_{\text{avg}} = \frac{34.044 - 29}{2.1 - 2} = \frac{5.044}{0.1} = 50.44$

2. Average velocity over $[2, 2.01]$:

• $t_2 = 2.01$
• $s(2.01) = 4(2.01)^3 - 3 = 4(8.120601) - 3 = 32.482404 - 3 = 29.482404$
• Average velocity:

$v_{\text{avg}} = \frac{29.482404 - 29}{2.01 - 2} = \frac{0.482404}{0.01} = 48.2404$

3. Average velocity over $[2, 2.001]$:

• $t_2 = 2.001$
• $s(2.001) = 4(2.001)^3 - 3 = 4(8.008006001) - 3 = 32.032024004 - 3 = 29.032024004$
• Average velocity:

$v_{\text{avg}} = \frac{29.032024004 - 29}{2.001 - 2} = \frac{0.032024004}{0.001} = 32.024$

4. Average velocity over $[2, 2.0001]$:

• $t_2 = 2.0001$
• $s(2.0001) = 4(2.0001)^3 - 3 = 4(8.00060006) - 3 = 32.00240024 - 3 = 29.00240024$
• Average velocity:

$v_{\text{avg}} = \frac{29.00240024 - 29}{2.0001 - 2} = \frac{0.00240024}{0.0001} = 24.0024$

Estimating the Instantaneous Velocity:

As the time intervals get smaller, the average velocities are converging to a value. Based on the values calculated, it appears that the instantaneous velocity $v_{\text{inst}}$ at $t = 2$ is approximately:

$v_{\text{inst}} \approx 48 \, \text{units/sec}$

This estimation is supported by the trend of the average velocities.

Would you like more details or have any questions?

Related Questions:

1. How do you find the derivative of a position function to determine instantaneous velocity?
2. What does the concept of limits have to do with instantaneous velocity?
3. How would the calculation change if the position function were different?
4. Can average velocity ever equal instantaneous velocity?
5. What are the practical applications of instantaneous velocity in physics?

Tip:

The instantaneous velocity at any point $t$ is simply the derivative of the position function $s(t)$ evaluated at that point.