To find the time taken by the particle to reach the velocity of $$60.0 \mathrm{~ms}^{-1}$$, we can use the formula:

$$ v = u + at $$

Where: $$v$$ is the final velocity, $$u$$ is the initial velocity, $$a$$ is the acceleration, and $$t$$ is the time taken.

Plugging in the given values:

$$ 60.0 \mathrm{~ms}^{-1} = 10.0 \mathrm{~ms}^{-1} + 2.0 \mathrm{~ms}^{-2} \cdot t $$

Solve for $$t$$:

$$ 50.0 \mathrm{~ms}^{-1} = 2.0 \mathrm{~ms}^{-2} \cdot t $$

$$ t = \frac{50.0 \mathrm{~ms}^{-1}}{2.0 \mathrm{~ms}^{-2}} = 25 \mathrm{s} $$

So, the time taken by the particle to reach the velocity of $$60.0 \mathrm{~ms}^{-1}$$ is 25 seconds.