I have this problem :
A 2.19-kg cart on a long, level, low-friction track is heading for a small electric fan at 0.21m/s . The fan, which was initially off, is turned on. As the fan speeds up, the magnitude of the force it exerts on the cart is given by $at^2 $ , where a = 0.0200 $N/s^2$.
I need to find the speed of the cart after 3.5 s of the fan being on and after how many seconds the velocity of the car will equal 0.
I'm not entirely sure what I should do to solve, I've tried drawing a free body diagram because that's the section this problem is from but that doesn't feel right. I've tried
$$\Sigma F=F(fan)=ma \space \\ \Sigma F=mg-N=ma=0 \\ v=v(initial)+at\\v=0.53$$
Which I am pretty sure is all wrong.
How is this suppose to be diagramed? Is this actually a free body diagram problem? If it is, how do I account for velocity since it isn't coming from the object. If it is not a free body diagram problem, do I just use general velocity equations to solve? I know conservation of mass and energy do not apply so I am not sure what else to do.