1. **Problem statement:** Calculate the curvature radius $\rho$ at time $t=2$ seconds given velocity $\mathbf{v}$ and acceleration $\mathbf{a}$. 2. The formula for curvature radius is: $$\rho = \frac{|\mathbf{v}|^3}{|\mathbf{v} \times \mathbf{a}|}$$ 3. Compute the cross product magnitude of velocity and acceleration vectors in 2D: $$|\mathbf{v} \times \mathbf{a}| = |v_x a_y - v_y a_x| = |2(-1) - (-2)(0)| = | -2 - 0| = 2$$ 4. Calculate the magnitude of velocity: $$|\mathbf{v}| = \sqrt{2^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.83$$ 5. Compute curvature radius using the given values: $$\rho = \frac{(2.83)^3}{2} = \frac{22.63}{2} = 11.31$$ 6. **Final answer:** The curvature radius at $t=2$ seconds is $\boxed{11.31}$ units.