**Average Velocity of a Particle with Uniform Acceleration**

To find the average velocity of a particle in the first distance, we need to consider the relationship between distance, time, initial velocity, acceleration, and average velocity.

**Motion Equation with Uniform Acceleration:**

The motion equation for a particle moving in a straight line with uniform acceleration is given by:

\[v = u + at\]

Where:
- \(v\) is the final velocity of the particle,
- \(u\) is the initial velocity of the particle,
- \(a\) is the uniform acceleration, and
- \(t\) is the time taken.

Since the initial velocity of the particle is given as 0, the equation simplifies to:

\[v = at\]

**Finding Average Velocity:**

The average velocity of a particle is defined as the total displacement divided by the total time taken. In this case, we want to find the average velocity in the first distance.

To determine the average velocity, we need to know the total distance traveled by the particle and the time taken to cover that distance. Since the initial velocity is 0, the total distance can be obtained by using the equation of motion for distance:

\[s = ut + \frac{1}{2}at^2\]

Where:
- \(s\) is the total distance traveled by the particle.

Since the initial velocity is 0, this equation simplifies to:

\[s = \frac{1}{2}at^2\]

We can rearrange this equation to solve for time:

\[t = \sqrt{\frac{2s}{a}}\]

Now that we have the time taken to cover the first distance, we can substitute it back into the equation for velocity:

\[v = a\sqrt{\frac{2s}{a}}\]

Simplifying further:

\[v = \sqrt{2as}\]

**Explanation:**

In summary, to find the average velocity of a particle in the first distance with a uniform acceleration, we need to use the motion equation for velocity and the equation for distance. By substituting the given initial velocity of 0 into the equation for velocity, we can find the time taken to cover the first distance. Then, by substituting this time into the equation for velocity, we can calculate the average velocity. The final equation for average velocity is \[v = \sqrt{2as}\].

Zero..