The distance traveled by an object in time $$t$$ is given by the equation $$s = (2.5)t^2$$. To find the instantaneous speed at a specific time, we need to find the first derivative of the distance function with respect to time, which gives us the velocity function:

$$v(t) = \frac{ds}{dt}$$

Differentiating the given equation with respect to $$t$$:

$$v(t) = \frac{d}{dt} (2.5)t^2 = 2(2.5)t = 5t$$

Now, we can find the instantaneous speed at $$t = 5 \mathrm{~s}$$ by plugging the value into the velocity function:

$$v(5) = 5(5) = 25 \mathrm{~m/s}$$

The instantaneous speed of the object at $$t = 5 \mathrm{~s}$$ is $$25 \mathrm{~m/s}$$.