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It looks like you've uploaded a math question involving position, velocity, and acceleration vectors from Section 4.1. Let me break down and solve the problem.

### Problem 1:
The position vector for a particle is given as a function of time by:
\[
\vec{r}(t) = x(t)\hat{i} + y(t)\hat{j}
\]
Where:
- \( x(t) = at + b \)
- \( y(t) = ct^2 + d \)

Given:
- \( a = 1.00 \, \text{m/s} \)
- \( b = 1.00 \, \text{m} \)
- \( c = 0.125 \, \text{m/s}^2 \)
- \( d = 1.00 \, \text{m} \)

We are asked to:
1. **(a)** Calculate the average velocity during the time interval \( t = 2.00 \, \text{s} \) to \( t = 4.00 \, \text{s} \).
2. **(b)** Determine the velocity and speed at \( t = 2.00 \, \text{s} \).

#### Part (a): Average Velocity
The average velocity is calculated using the formula:
\[
\vec{v}_{\text{avg}} = \frac{\vec{r}(t_2) - \vec{r}(t_1)}{t_2 - t_1}
\]
Where:
- \( t_1 = 2.00 \, \text{s} \)
- \( t_2 = 4.00 \, \text{s} \)

We will calculate \( \vec{r}(t_2) \) and \( \vec{r}(t_1) \), then find the difference and divide by the time interval.

#### Part (b): Velocity at \( t = 2.00 \, \text{s} \)
The velocity vector is the time derivative of the position vector:
\[
\vec{v}(t) = \frac{d\vec{r}}{dt} = \frac{dx}{dt}\hat{i} + \frac{dy}{dt}\hat{j}
\]
The speed is the magnitude of the velocity vector:
\[
v = \sqrt{v_x^2 + v_y^2}
\]

Let me calculate the values now.### Solution:

#### Part (a): Average Velocity
The average velocity between \( t = 2.00 \, \text{s} \) and \( t = 4.00 \, \text{s} \) is:
\[
\vec{v}_{\text{avg}} = (1.00 \, \hat{i} + 0.75 \, \hat{j}) \, \text{m/s}
\]

#### Part (b): Velocity and Speed at \( t = 2.00 \, \text{s} \)
- The velocity at \( t = 2.00 \, \text{s} \) is:
\[
\vec{v}(t = 2.00 \, \text{s}) = 1.00 \, \hat{i} + 0.50 \, \hat{j} \, \text{m/s}
\]
- The speed (magnitude of velocity) at \( t = 2.00 \, \text{s} \) is:
\[
\text{Speed} = 1.12 \, \text{m/s}
\]

Would you like further clarification on any part of the solution?

### Related Questions:
1. How is average velocity different from instantaneous velocity?
2. What is the physical meaning of the velocity vector's components?
3. How can the acceleration vector be determined from the given position functions?
4. How does changing the constant \( a \) affect the velocity and speed at a given time?
5. Can the average speed be different from the magnitude of the average velocity?

#### Tip: 
For two-dimensional motion, always remember that speed is a scalar (magnitude only), while velocity is a vector (magnitude and direction).

Suppose the position vector for a particle is given as a function of time by r(t) = x(t)i + y(t)j, with x(t) = at + b and y(t) = ct^2 + d, where a = 1.00 m/s, b = 1.00 m, c = 0.125 m/s^2, and d = 1.00 m. (a) Calculate the average velocity during the time interval from t = 2.00 s to t = 4.00 s. (b) Determine the velocity and speed at t = 2.00 s.

Solve a problem involving the position vector of a particle as a function of time, given by r(t) = x(t)i + y(t)j. Learn how to calculate average velocity over a time interval and determine the instantaneous velocity and speed at a specific time.

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