Velocity(t), of rod moving in B-field

[serverxeon]

Here's the question

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Here's my attempt using energy approach.

However, I'm kinda stuck. how do I solve this DE to get the expression for v(t)?

 

[TSny]

I think it will be much simpler to just apply Newton's second law: $F = ma = m\dot{v}$

[You could take the derivative with repsect to t of both sides of your last equation and get a differential equation for v(t), but it will be the same as just setting up F = ma.]