A uniform magnetic field ` vecB` is set up along the positive x-axis. A particle of charge 'q' and mass 'm' moving with a velocity `vecv` entresthe field at the origin in X-Y plane such that it has velocity companents both along and perpendicular to the magnetic field `vecB`. Trace, giving reason. the fraectory followed by the particle . Find out the expression for the distance moved by the particle along the magnetic field in one rotation.

Explanation

To solve the problem, we need to analyze the motion of a charged particle entering a uniform magnetic field. The magnetic field is directed along the positive x-axis, and the particle has both parallel and perpendicular components of velocity relative to the magnetic field.

### Step-by-Step Solution:

1. **Understanding the Motion**:
- The magnetic field \( \vec{B} \) is directed along the x-axis.
- The particle has a charge \( q \) and mass \( m \), and it enters the magnetic field with a velocity \( \vec{v} \) at the origin in the x-y plane.
- The velocity can be resolved into two components:
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