A particle of specific charge `alpha` is projected from origin with velocity `v=v_0hati-v_0hatk` in a uniform magnetic field `B=-B_0hatk`. Find time dependence of velocity and position of the particle.

Explanation

To solve the problem, we need to analyze the motion of a charged particle in a magnetic field. The particle has a specific charge \( \alpha \) and is projected from the origin with an initial velocity \( \mathbf{v} = v_0 \hat{i} - v_0 \hat{k} \) in a uniform magnetic field \( \mathbf{B} = -B_0 \hat{k} \).

### Step 1: Understanding the Motion
The magnetic field \( \mathbf{B} \) is directed along the negative z-axis, while the initial velocity has components in the x and z directions. The y-component of the velocity is zero initially. The magnetic force will act on the particle, causing it to move in a circular path in the xy-plane, while the z-component of the velocity will remain unchanged.

### Step 2: Magnetic Force Calculation
The magnetic force \( \mathbf{F} \) acting on the particle is given by the Lorentz force equation:
\[
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