A long, tightly-wound solenoid of radius r has n turns per unit...

The magnetic field inside a long solenoid is uniform and directed along its axis.

$B = \mu_0 n i$

When a particle of charge $q$ and mass $m$ is projected from the axis perpendicular to the magnetic field with speed $v$, it moves in a circular path. The radius of this circular path is given by:

$R = \dfrac{mv}{qB}$

Since the particle is projected from the axis, the axis lies on the circumference of its circular trajectory. The maximum distance of the particle from the axis is equal to the diameter of its circular path, which is $2R$.

For the particle to not strike the solenoid, this maximum distance must be less than or equal to the radius of the solenoid $r$.

$2R \le r$

$2 \left( \dfrac{mv}{qB} \right) \le r$

$v \le \dfrac{qBr}{2m}$

Substituting the value of the magnetic field $B = \mu_0 n i$:

v_\max = \dfrac{\mu_0 q r n i}{2m}