A particle having mass m and charge `q` is released from the origin in a region in which ele field and magnetic field are given by `B=-B_0hatj`and `E=E_0hatk`.
Find the y- component of the velocity and the speed of the particle as a function of it z-coordinate.

Explanation

To solve the problem step-by-step, we will analyze the motion of a charged particle in the presence of electric and magnetic fields.

### Step 1: Identify the Forces Acting on the Particle
The particle has a charge \( q \) and is subject to an electric field \( \mathbf{E} = E_0 \hat{k} \) and a magnetic field \( \mathbf{B} = -B_0 \hat{j} \). The forces acting on the particle are given by:
- The electric force:
\[ \mathbf{F}_E = q \mathbf{E} = q E_0 \hat{k} \]
- The magnetic force (using the Lorentz force law):
\[ \mathbf{F}_B = q (\mathbf{v} \times \mathbf{B}) \]

### Step 2: Determine the Velocity Components
Initially, the particle is released from the origin, so its initial velocity \( \mathbf{v} = 0 \). As it starts moving, we need to find the y-component of the velocity \( v_y \).

Since the magnetic field is in the negative y-direction and the electric field is in the positive z-direction, the magnetic force will act perpendicular to the velocity. However, at the moment of release, the velocity is zero, which means:
\[ v_y(0) = 0 \]

### Step 3: Analyze the Motion in the z-Direction
The electric field will exert a force in the z-direction, causing the particle to accelerate in that direction. The work done by the electric field will change the kinetic energy of the particle.

Using the work-energy theorem, the work done by the electric field on the particle as it moves a distance \( z \) in the z-direction is:
\[ W = q E_0 z \]
This work done will be equal to the change in kinetic energy:
\[ \frac{1}{2} m v^2 = q E_0 z \]
From this, we can solve for the speed \( v \):
\[ v = \sqrt{\frac{2 q E_0 z}{m}} \]

### Step 4: Relate the Velocity Components
Since the particle is only influenced by the electric field in the z-direction, and there is no initial velocity in the y-direction, the y-component of the velocity remains zero:
\[ v_y = 0 \]

### Final Results
Thus, the y-component of the velocity is:
\[ v_y = 0 \]
And the speed of the particle as a function of its z-coordinate is:
\[ v = \sqrt{\frac{2 q E_0 z}{m}} \]