Equation of motion for positive charge injected into uniform magnetic field

[big man]

OK first off here is the problem.

problem: A positive point charge q of mass m is injected with a velocity $$u_0 = \mu_0 j$$ into the y > 0 region where a uniform magnetic field $$B = B_0 i$$ exists. Obtain the equation of motion of the charge, and describe the path that the charge follows.

i, j and k represent unit vectors in the direction of the x, y and z axes.

Now I understand that the path will be a semicircle from the theory of the motion of a charged particle in a uniform magnetic field and since y > 0 and I know that the force experienced by the particle will be in the z direction.

$$F_m = q(u_0 X B) = q \mu_0 B_0 k$$

The velocity is going to be constant, but the motion will vary with time in the z direction I think? Obviously my problem is that I'm not too sure of how to solve for the equation of motion in this problem. I guess one of the reasons why I'm finding this so difficult is I can't even visualise it properly.

Any help to put me on the right track would be great.

Thanks

EDIT: I can get the answer by simply equating the centripetal and magnetic forces and then substituting $$R= \frac {m \mu_0} {q B_0}$$ into the general equation of a semicircle, but I don't think this is how they want you to do it.

 

[Astronuc]

$$u_0 = \mu_0 j$$ That's the initial velocity when it enters the magnetic field.

The velocity does not remain in the j-direction though. The particle accelerates, i.e. changes direction.

If the 'speed' is constant, what can one say about the trajectory?

If it leaves the magnetic field, it will travel in a straight line in the direction of the velocity vector where it leaves the M field, unless subject some other E or M field, not parallel with the velocity.