To solve the problem, we'll break it down into three parts:

1. Find the velocity at $t = 0$ seconds.
2. Find the velocity at $t = 4$ seconds.
3. Find the average velocity between $t = 2$ seconds and $t = 4$ seconds.

1. Velocity at $t = 0$ seconds

The velocity of the object is given by the derivative of the position function with respect to time:

$v(t) = \frac{dy}{dt}$

Given $y(t) = a + bt^2$, where $a = 8$ m and $b = 2$ m:

$v(t) = \frac{d}{dt}(8 + 2t^2)$

Taking the derivative:

$v(t) = 0 + 2 \cdot 2t = 4t$

At $t = 0$ seconds:

$v(0) = 4 \cdot 0 = 0 \, \text{m/s}$

2. Velocity at $t = 4$ seconds

Using the same velocity equation $v(t) = 4t$:

$v(4) = 4 \cdot 4 = 16 \, \text{m/s}$

3. Average Velocity between $t = 2$ seconds and $t = 4$ seconds

The average velocity between two times is given by:

$\text{Average Velocity} = \frac{y(t_2) - y(t_1)}{t_2 - t_1}$

Here, $t_1 = 2$ seconds and $t_2 = 4$ seconds.

First, calculate $y(t_2) = y(4)$ and $y(t_1) = y(2)$:

$y(t) = 8 + 2t^2$

$y(4) = 8 + 2(4)^2 = 8 + 2 \cdot 16 = 8 + 32 = 40 \, \text{m}$

$y(2) = 8 + 2(2)^2 = 8 + 2 \cdot 4 = 8 + 8 = 16 \, \text{m}$

Now, calculate the average velocity:

$\text{Average Velocity} = \frac{y(4) - y(2)}{4 - 2} = \frac{40 - 16}{2} = \frac{24}{2} = 12 \, \text{m/s}$

Summary:

• Velocity at $t = 0$ seconds: $0$ m/s
• Velocity at $t = 4$ seconds: $16$ m/s
• Average velocity between $t = 2$ seconds and $t = 4$ seconds: $12$ m/s

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:

1. How would the velocity change if $b$ were negative?
2. What would the acceleration of the object be at any time $t$?
3. How does the position function change if $a$ is altered?
4. Can you find the displacement between $t = 2$ and $t = 4$ seconds?
5. How would the average velocity change if the time interval were between $t = 0$ and $t = 4$ seconds?

Tip: The derivative of a position function gives the velocity function, and the derivative of the velocity function gives the acceleration function.