Constant acceleration average velocity is a concept in physics that describes the average speed of an object moving with constant acceleration over a given time interval. It is particularly useful in analyzing motion where the velocity changes at a constant rate, such as in free fall or uniformly accelerated motion.

The formula for average velocity under constant acceleration is:

\[ \bar{v} = \frac{v_0 + v}{2} \]

Where:

- \(\bar{v}\) is the average velocity
- \(v_0\) is the initial velocity
- \(v\) is the final velocity

This formula can also be expressed in terms of initial velocity, acceleration, and time:

\[ \bar{v} = v_0 + \frac{1}{2}at \]

Where:

- \(a\) is the acceleration
- \(t\) is the time interval

Let's calculate the average velocity of an object that starts with an initial velocity of 5 m/s and accelerates at 2 m/s² for 10 seconds:

- Identify the known values:
- Initial velocity (\(v_0\)) = 5 m/s
- Acceleration (\(a\)) = 2 m/s²
- Time (\(t\)) = 10 s

- Apply the average velocity formula: \[ \bar{v} = v_0 + \frac{1}{2}at \]
- Substitute the known values: \[ \bar{v} = 5 \text{ m/s} + \frac{1}{2}(2 \text{ m/s²})(10 \text{ s}) \]
- Perform the calculation: \[ \bar{v} = 5 \text{ m/s} + 10 \text{ m/s} = 15 \text{ m/s} \]

Let's visualize the constant acceleration average velocity with our example:

This visual representation shows:

- The x-axis represents time, from 0 to 10 seconds
- The y-axis represents velocity, from 0 to 25 m/s
- The green line shows the linear increase in velocity over time
- The red dashed line represents the average velocity (15 m/s)
- The yellow point indicates the initial velocity (5 m/s) at t = 0 s
- The blue point indicates the final velocity (25 m/s) at t = 10 s

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