A paricle is projected up from the bottom of an inlined plane of inclination `alpha` with velocity `v_0` If it returns to the points of projection after an elastic impact with the plane, find the total time of motion of the particle.

Explanation

To solve the problem of a particle projected up an inclined plane and returning to its original position after an elastic impact, we can follow these steps:

### Step 1: Understand the Motion
The particle is projected with an initial velocity \( v_0 \) at an angle \( \theta \) to the inclined plane, which is at an angle \( \alpha \) to the horizontal. The motion can be analyzed in two dimensions: along the inclined plane (x-direction) and perpendicular to the inclined plane (y-direction).

### Step 2: Resolve Initial Velocity
We resolve the initial velocity \( v_0 \) into components:
- Along the inclined plane (x-direction):
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