(a) Show that the velocity acquired by a particle in sliding down an inclined plane is the same as that acquired by a particle falling freely from frst though a distance equal to the height of the inclined plane. (b) Find the time taken in sliding a particle down the whole length of the incline.

Explanation

To solve the problem, we will break it down into two parts as given in the question.

### Part (a): Show that the velocity acquired by a particle in sliding down an inclined plane is the same as that acquired by a particle falling freely from a height equal to the height of the inclined plane.

1. **Identify the height (h)**: Let the height of the inclined plane be \( h \) and the angle of inclination be \( \theta \).

2. **Free fall case**: For a particle falling freely from a height \( h \), we can use the principle of conservation of energy. The potential energy at the height \( h \) is converted into kinetic energy just before it hits the ground.
- Initial potential energy: \( PE = mgh \)
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