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# How do you simplify \frac{at^{2}}{2}+ut?

Nov 1, 2017

$\frac{a {t}^{2}}{2} + u t = \overline{v} \times t$, where $\overline{v}$ is average velocity.

#### Explanation:

We can write $\frac{a {t}^{2}}{2} + u t$ as $\frac{a {t}^{2} + 2 u t}{2}$, if you mean so the question has been answered.

This represents the distance $S$ covered an object with an initial velocity $u$ and a uniform accelaration $a$ in time $t$ i.e. $S = \frac{a {t}^{2}}{2} + u t$ or $S = \frac{a {t}^{2} + 2 u t}{2}$.

It may be noted that if initial velocity is $u$ and uniform accelaration is $a$, then the velocity after $t$ units of time $v$ is given by

$v = u + a t$ and average velocity $\overline{v}$ is given by $\overline{v} = \frac{v + u}{2}$ i.e.

but as $S = \frac{a {t}^{2} + 2 u t}{2} = \frac{t \left(a t + u + u\right)}{2} = \frac{t \left(v + u\right)}{2}$

or we can say $S = \overline{v} \times t$, where $\overline{v}$ is average velocity.

Note that the different formulas are used in different circumstances and though $\frac{a {t}^{2}}{2} + u t$ simplifies to $\overline{v} \times t$, these are used depending on what variables are given to us, regarding motion with uniform accelaration. One more important formula is ${v}^{2} - {u}^{2} = 2 a S$.

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