I can't think of the meaning of squaring the Time (multiplying it by itself). It makes sense in Mathematics. But how can you figure it out in nature (or physics). As an example, the formula $$s=ut+(1/2)at^2.$$
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Well, think of it this way: Let's say you have a quantity that represents change over time. Could be velocity (distance travelled per unit of time), could be precipitation (inches of rain falling per hour), could be flow rate (liters per unit of time).
Now if you want to know how much stuff you get (distance travelled, inches of rain fallen, liters of water flown) after a certain amount of time, you multiply the rate with the time.
So, total distance travelled is velocity multiplied with time, total rainfall amount is rain fall rate multiplied with time, etc. etc.
Now, what if the thing that's changing over time also is something that depends on time? The easiest example here is acceleration. Acceleration tells you how much the velocity changes over time. For a car, you'd advertise how fast it can go from 0 to 100 km/h.
So, if you multiply acceleration with time, you get the resulting velocity, and to get a distance, you multiply that with time again. That's why you have "time squared"