The position of an object moving along the y-axis is given by y = a + bt², where a = 8m, b = 2m and t is measured in seconds. What is its velocity at t = 0s and t = 4s? What is the average velocity between t = 4s and t = 2s?

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.

Learn how to calculate velocity and average velocity of an object moving along the y-axis using calculus. This problem involves finding velocity at specific times and calculating average velocity over a time interval. Suitable for high school students.

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