Problem

The key insight is that the magnetic force $\mathbf{F} = q\mathbf{v} \times \mathbf{B}$ depends on position through the field strength, so the particle won't follow a simple circular arc. Instead, use energy conservation—the magnetic force does no work, so kinetic energy stays constant—combined with the fact that the particle reaches maximum $x$ when its $x$-component of velocity becomes zero. At that turning point, all the initial momentum in the $x$-direction has been redirected by the cumulative magnetic deflection.

The key insight is that the magnetic force $\mathbf{F} = q\mathbf{v} \times \mathbf{B}$ depends on position through the field strength, so the particle won't follow a simple circular arc. Instead, use energy conservation—the magnetic force does no work, so kinetic energy stays constant—combined with the fact that the particle reaches maximum $x$ when its $x$-component of velocity becomes zero. At that turning point, all the initial momentum in the $x$-direction has been redirected by the cumulative magnetic deflection.