The Equations of Uniform Acceleration

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The SUVAT Equations (in no particular order)	Missing Variable
$$v=u+at$$	$s$
$$s=vt-\frac{1}{2}a{t}^2$$	$u$
$$s=ut+\frac{1}{2}a{t}^2$$	$v$
$$s=\frac{u+v}{2}t$$	$a$
$$v^2=u^2+2as$$	$t$

A table of the rearrangement of these equations.

Each variable has the following meanings

$s$	Displacement	metres / $m$
$u$	Initial velocity	metres per second / $m{s}^{-1}$
$v$	Final velocity	metres per second / $m{s}^{-1}$
$a$	Acceleration	metres per second squared / $m{s}^{-2}$
$t$	Time	seconds / $s$

Further details can be found at Wikipedia: Equations of motion - Uniform acceleration

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SUVAT Questions
(Care of mmerevise.co.uk)

More Notes and Questions
(Care of www.gcsescience.com)

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$a=\frac{v-u}{t}$

Definition of Acceleration
(or s-missing)

$$a=\frac{v-u}{t}$$

where the variables have the usual SUVAT meanings

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Starting from the definition of acceleration and the idea of constant acceleration

$$a=\frac{\Delta v}{\Delta t}$$

where $\Delta v$ is the change in velocity and $\Delta t$ is the time taken for this change.

$\Delta v$ is the final velocity minus the initial, i.e., $\Delta v=v-u$. So the acceleration becomes

$$a=\frac{v-u}{t}$$

where we have replace $\Delta t$ with just $t$

Taking the acceleration as constant then this equation can be written:

$$v=u+at$$

As required

For those used to calculus we can use integration. Taking,

$$a=\frac{\Delta v}{\Delta t}$$

to the limit, we get,

$$a=\underset{t\to 0}{\mathrm{Lim}}\frac{\Delta v}{\Delta t}=\frac{dv}{dt}$$

integrating with respect to $t$

$$\int adt=v$$ $$at+C=v$$

where $C$ is a constant of integration and can be found by knowing $v=u$ when $t=0$.

$$at+u=v$$

as before.

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Further Reading:

$$s=ut+\frac{1}{2}a{t}^2$$

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$s=vt-\frac{1}{2}a{t}^2$

(u-missing)

Prerequesite materials:

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Derivation:

Rearranging the first of the above equations:

$$u=v-at$$

Substituting this into the second equation:

$$s=(v-at)t+\frac{1}{2}a{t}^2$$ $$s=vt-a{t}^2+\frac{1}{2}a{t}^2$$ $$s=vt-\frac{1}{2}a{t}^2$$

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The average speed equation
$\frac{s}{t}=\frac{v+u}{2}$
(or $a$-missing)

Prerequisite Materials

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Describing the average speed equations:

$$\frac{s}{t}=\frac{v+u}{2}$$

where the variables have the usual SUVAT meanings

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Derivation of a-missing:

Rearranging each of the prerequisite equations to make $a$ the subject gives

Equating

$$\frac{v-u}{t}=\frac{v^2-u^2}{2s}$$ $$\frac{v-u}{t}=\frac{(v+u)(v-u)}{2s}$$ $$\frac{1}{t}=\frac{v+u}{2s}$$ $$\frac{s}{t}=\frac{v+u}{2}$$

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