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Let's discuss a beautiful problem from Physics Olympiad based on Velocity and Acceleration.

**The Problem: Velocity and Acceleration**

A particle is moving in positive x-direction with its velocity varying as v= α√x. Assume that at t=0, the particle was located at x=0. Determine the

- the time dependence of velocity
- acceleration
- the mean velocity of the particle averaged over the time that the particle takes to cover the first s metres of the path.

**Discussion:**

v=α√x.

Squaring both sides

v^{2}=α^{2}x

=2(α^{2}/2)x

Acceleration= α^{2}/2

The initial velocity u is therefore zero and the acceleration is constant.

- v=u+at= (α
^{2}/2)t

- The acceleration= α
^{2}/2 - V=α√s

Average velocity=(0+v)/2=α√s/2

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