# Why there is a $\frac{1}{2}$ in the distance formula $d=\frac{1}{2}at^2$?

I'm preparing for my exam, but I have difficulties in perceiving why there is a $\frac{1}{2}$ in the distance formula $d=\frac{1}{2}at^2$?

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Integrate acceleration twice wrt time. – jinawee Dec 9 '13 at 11:51

It is exactly because we have a factor of $\frac 1 2$ in the area formula of a triangle. To understand what I'm saying, consider what is the $v(t)$ graph of a particle under constant acceleration.

Some say, a good plot is worth a million words! :)

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+1: Great and simple to comprehend answer. – user29727 Dec 9 '13 at 13:05
The positive reaction to this answer is way better than what I was expecting! Thanks to everyone. – Ali Dec 9 '13 at 17:54
@Ali Part of it I think is that clarity in technical writing is a jewel to behold. – WetSavannaAnimal aka Rod Vance Dec 10 '13 at 13:36

We can also do it using calculus,the displacement of the particle is given by

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$\Delta x =v_{average}\times t$

In uniform acceleration $v_{average}$ becomes $\dfrac{v-v_{0}}{2}$

Hence;

$\Delta x =\dfrac{v-v_{0}}{2}t=\dfrac{1}{2}\dfrac{v-v_{0}}{t}t^{2}=\dfrac{1}{2}at^{2}$

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## protected by Qmechanic♦Dec 17 '13 at 9:04

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