$v^2=u^2+2as$
The Energy Equation
(t-missing)

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Prerequesite Materials:

Describing "the Energy Equation"

$$v^2=u^2+2as$$

Derivation:

Taking

$$v=u+at$$

and rearranging for $t$

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

Then substitute this rearrangement into

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

To give

$$s=\left(\frac{u+v}{2}\right)\left(\frac{v-u}{a}\right)$$

Simplifying

$$s=\frac{v^2-u^2}{2a}$$ $$v^2=u^2+2as$$

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We can also derived this directly from the following using calculus.

$$\frac{dv}{dt}=a$$

Using the chain rule

$$\frac{dv}{ds}\frac{ds}{dt}=a$$ $$\frac{dv}{ds}v=a$$

Integrating with respect to $s$

$$\int \frac{dv}{ds}vds=a\int ds$$ $$\int vdv=a\int ds$$ $$\frac{v^2}{2}=as+C$$ $$v^2=2as+C$$

$v=u$ when $s=0$. So,

$$u^2=2a\times 0+C$$ $$C=u^2$$

Therefore,

$$v^2=2as+u^2$$

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Why "the Energy Equation"?

$$v^2=u^2+2as$$

Multiply this by the mass, $m$ of the moving object, we get

$$m{v}^2=m{u}^2+2mas$$

Noting that from Newton's Second Law, $ma=F$ then

$$m{v}^2=m{u}^2+2Fs$$

Using the definition of Energy, $W=Fs$ then

$$m{v}^2=m{u}^2+2W$$

Then rearranging to make $W$ the subject:

$$\frac{1}{2}m{v}^2=\frac{1}{2}m{u}^2+W$$ $$W=\frac{1}{2}m{v}^2-\frac{1}{2}m{u}^2$$

This is the Work-Energy Principle which states that the work done on an object is equal to the change in kinetic energy.

$$\text{Work Done}=\text{Change in Kinetic Energy}$$ $$\text{W}=\Delta E_{\text{kin}}$$

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