Blurring transfer function of image
Your start looks right, although I think your speed equation is wrong. To continue, the FT has no time dependance so you can take the integral out.

$$G(u, v) = \int_0^T \mathcal F\{f(x-x_0, y-y_0)\} dt $$

$$ = F(u,v) \int_0^T e^{-2\pi i(x_0u + y_0v)} dt$$

Divide by $F$ to get the transfer function.

If I understand the question correctly $x_0(t)$ and $y_0(t)$ are positions such that $x_0(t) = x_0(0) + v_x(t)t$ and similar for $y$. You can then put in an appropriate version of the speed given and do the integration.