What is vectorr(t) when the acceleration is 3m/s^2

To find the vector **r(t)** (position vector as a function of time), we need to relate it to the acceleration vector. Given that acceleration **a(t)** is constant at \( 3 \ \text{m/s}^2 \), we can integrate to find the velocity vector **v(t)** and then integrate again to find the position vector **r(t)**.

### Step-by-step solution:

1. **Acceleration** \( a(t) \):
   \[
   a(t) = 3 \ \text{m/s}^2
   \]
   This implies that acceleration is constant.

2. **Velocity** \( v(t) \):
   To find velocity, integrate acceleration with respect to time:
   \[
   v(t) = \int a(t) \, dt = \int 3 \, dt = 3t + C_1
   \]
   where \( C_1 \) is the constant of integration, representing the initial velocity \( v_0 \).

3. **Position** \( r(t) \):
   Now, integrate the velocity function with respect to time:
   \[
   r(t) = \int v(t) \, dt = \int (3t + C_1) \, dt = \frac{3t^2}{2} + C_1 t + C_2
   \]
   where \( C_2 \) is another constant of integration, representing the initial position \( r_0 \).

### Final expression for \( r(t) \):
\[
r(t) = \frac{3t^2}{2} + v_0 t + r_0
\]
Here:
- \( v_0 \) is the initial velocity at \( t = 0 \),
- \( r_0 \) is the initial position at \( t = 0 \).

To fully determine \( r(t) \), you would need the initial conditions: \( v_0 \) and \( r_0 \).

Would you like more details on this? Or perhaps clarification on the integration process?

### Related Questions:
1. How do you find the velocity when the acceleration is non-constant?
2. What are the steps for solving kinematic problems with initial velocity and acceleration?
3. How do you interpret the constants \( v_0 \) and \( r_0 \) in real-world problems?
4. How would the result change if acceleration was a vector instead of a scalar?
5. How do you graph position vs. time when acceleration is constant?

### Tip:
When solving kinematic problems, always check for initial conditions like starting velocity and position to fully determine the motion.

What is vector r(t) when the acceleration is 3m/s^2

This problem shows how to determine the position vector r(t) when the acceleration is constant at 3 m/s^2. Using kinematic equations and integration, learn how to derive velocity and position vectors with step-by-step solutions, suitable for high school and college-level students.

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