A charge in motion

Relevant wiki: Lorentz Force Law (Magnetic and Mixed Fields)

$\text{The max } x \text{ coordinate will be reached when the particle has no velocity in }x \text{ coordinate, i.e }{ v }_{ x }=0 \text{ and } { v }_{ y }={ v }_{ 0 }$ . $\text{The direction of force is determined using Fleming's left hand rule, which will come out in }+y \text{ direction.}$ $\text{Even if the angle between } v \text{ and } B \text{ is 90 degrees, the particle wont undergo circular motion as the magnetic field is not constant.}$

$\text{ Now, using Lorentz force equation we get }$ $m{ a }_{ y }=q{ v }_{ 0 }{ B }_{ 0 }x cos{ \theta }\quad \Rightarrow \quad m{ a }_{ y }=q({ v }_{ 0 }cos{ \theta })B_{ 0 }x\\ m\dfrac { d{ v }_{ y } }{ dt } =q{ B }_{ 0 }x\dfrac { dx }{ dt } \quad \Rightarrow m\int _{ 0 }^{ { v }_{ 0 } }{ d } { v }_{ y }=q{ B }_{ 0 }\displaystyle\int _{ 0 }^{ { x }_{ max } }{ xdx } \\ m{ v }_{ 0 }-0=q{ B }_{ 0 }\dfrac { { x }_{ max }^{ 2 } }{ 2 } \quad \Rightarrow { x }_{ max }=\sqrt { \dfrac { 2m{ v }_{ 0 } }{ q { B }_{ 0 }} }$