A particle retards from a velocity `v_(0)`while moving in a straight line. If the magnitude of deceleration is directly proportional to the square loop of the speed of the particle, find its average velocity for the total time of its motion.

Explanation

To solve the problem, we need to find the average velocity of a particle that is retarding from an initial velocity \( v_0 \) under the condition that the deceleration is directly proportional to the square root of the speed of the particle.

### Step-by-Step Solution:

1. **Understanding the Deceleration**:
The problem states that the deceleration \( a \) is proportional to the square root of the speed \( v \). We can express this mathematically as:
\[
a = -\alpha \sqrt{v}
...