The task is to find the average velocities over two time intervals and then estimate the instantaneous velocity of the car at $t = 0.2$ seconds.

Part (a)

To find the average velocity over the interval $0 \leq t \leq 0.2$ seconds: The formula for average velocity is given by: $\text{Average Velocity} = \frac{\Delta s}{\Delta t}$ where $\Delta s$ is the change in position and $\Delta t$ is the change in time.

From the table:

• $s(0) = 0 \, \text{ft}$
• $s(0.2) = 0.5 \, \text{ft}$
• $\Delta s = 0.5 - 0 \, \text{ft} = 0.5 \, \text{ft}$
• $\Delta t = 0.2 - 0 \, \text{sec} = 0.2 \, \text{sec}$

Plugging in the values: $\text{Average Velocity} = \frac{0.5 \, \text{ft}}{0.2 \, \text{sec}} = 2.5 \, \text{ft/sec}$

Part (b)

To find the average velocity over the interval $0.2 \leq t \leq 0.4$ seconds:

• $s(0.2) = 0.5 \, \text{ft}$
• $s(0.4) = 1.7 \, \text{ft}$
• $\Delta s = 1.7 - 0.5 \, \text{ft} = 1.2 \, \text{ft}$
• $\Delta t = 0.4 - 0.2 \, \text{sec} = 0.2 \, \text{sec}$

Plugging in the values: $\text{Average Velocity} = \frac{1.2 \, \text{ft}}{0.2 \, \text{sec}} = 6 \, \text{ft/sec}$

Part (c)

To estimate the instantaneous velocity at $t = 0.2$ seconds, you can use the average velocities obtained from parts (a) and (b). Since the instantaneous velocity at a point is the limit of the average velocity as the interval shrinks around the point, we can average the two average velocities: $\text{Instantaneous Velocity at } t = 0.2 = \frac{2.5 + 6}{2} = 4.25 \, \text{ft/sec}$

Would you like to explore more about the calculations or have any other questions on this topic? Here are some options you might consider:

1. How does the average velocity compare to the instantaneous velocity, and why might they differ?
2. What would be the estimated instantaneous velocity at $t = 0.4$ seconds?
3. How do you calculate the average acceleration of the car over the interval $0 \leq t \leq 0.4$?
4. How might the velocity function of the car look like based on these average velocities?
5. Could the actual path of the car be linear or non-linear based on the velocities calculated?

Tip: To find the instantaneous velocity more accurately, consider using smaller time intervals if the data is available. This approximates the derivative at a point more closely.