An object is moving in the xy plane. At t=1s, the object’s velocity is 3m/s in the +x direction. At t=2s, the object’s velocity is 4 m/s in the +y direction. What is the average acceleration of the object between t=1s and t=2s?

The average acceleration of an object is given by the formula:

\[
\vec{a}_{\text{avg}} = \frac{\Delta \vec{v}}{\Delta t}
\]

where:
- \(\Delta \vec{v}\) is the change in velocity,
- \(\Delta t\) is the change in time.

### Step 1: Determine the initial and final velocities
At \( t = 1 \, \text{s} \):
- The velocity is \( \vec{v}_1 = 3 \, \text{m/s} \) in the \( +x \)-direction.
  \[
  \vec{v}_1 = 3 \hat{i} \, \text{m/s}
  \]

At \( t = 2 \, \text{s} \):
- The velocity is \( \vec{v}_2 = 4 \, \text{m/s} \) in the \( +y \)-direction.
  \[
  \vec{v}_2 = 4 \hat{j} \, \text{m/s}
  \]

### Step 2: Calculate the change in velocity
The change in velocity is:

\[
\Delta \vec{v} = \vec{v}_2 - \vec{v}_1 = (0 \hat{i} + 4 \hat{j}) - (3 \hat{i} + 0 \hat{j})
\]

\[
\Delta \vec{v} = -3 \hat{i} + 4 \hat{j} \, \text{m/s}
\]

### Step 3: Calculate the average acceleration
The time interval is:
\[
\Delta t = t_2 - t_1 = 2 \, \text{s} - 1 \, \text{s} = 1 \, \text{s}
\]

Now, calculate the average acceleration:
\[
\vec{a}_{\text{avg}} = \frac{\Delta \vec{v}}{\Delta t} = \frac{-3 \hat{i} + 4 \hat{j}}{1} = -3 \hat{i} + 4 \hat{j} \, \text{m/s}^2
\]

Thus, the average acceleration of the object is:
\[
\vec{a}_{\text{avg}} = -3 \hat{i} + 4 \hat{j} \, \text{m/s}^2
\]

This means the object is accelerating at \( 3 \, \text{m/s}^2 \) in the negative \( x \)-direction and \( 4 \, \text{m/s}^2 \) in the positive \( y \)-direction.

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Would you like more details, or do you have any questions?

### Follow-up Questions:
1. How would you calculate the magnitude of the average acceleration?
2. What is the direction of the average acceleration in terms of angles?
3. Can the motion be described by any particular type of curve in the xy-plane?
4. How would the acceleration change if the velocities at different times were given?
5. How would you describe the motion in terms of kinematic equations?

### Tip:
Remember that velocity and acceleration are vector quantities, so direction matters when calculating changes.

Find the average acceleration of an object moving in the xy plane, with velocity 3 m/s in the +x direction at t=1s and 4 m/s in the +y direction at t=2s. Learn how to calculate acceleration using vector subtraction and kinematics, suitable for students in Grades 9-12.

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