ATAR Notes: Forum

VCE Stuff => VCE Science => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Physics => Topic started by: Mao on January 04, 2008, 09:22:51 am

Title: Deriving Ek
Post by: Mao on January 04, 2008, 09:22:51 am: I was just wondering the other day, many of the VCE physics formulas are derived from very simple equations, e.g. the 5 formulas dealing with constant acceleration and etc...

however no textbooks i've come across to have shown the workings to $E_k=\frac{1}{2} \cdot m \cdot v^2$
obviously you do not need that for the course, but just out of curiosity, where is this formula derived from? if at all??

i tried substituting stuff to $w=f \cdot d$ , but most of it ended nowhere...
this was what i got...

$w=f \cdot d$

given that $v^2=u^2+2 \cdot a \cdot d \Rightarrow d= \frac{v^2-u^2}{2 \cdot a}$

therefore $w=E_k=m \cdot a \cdot \left( \frac{v^2-u^2}{2 \cdot a} \right)$

$\Rightarrow E_k=\frac{1}{2} \cdot m \cdot \left( v^2-u^2 \right)$ given that $a \neq 0$

and i'm stuck....
:(

Thanks!!
Title: Re: Deriving Ek
Post by: Collin Li on January 04, 2008, 09:31:50 am: http://en.wikipedia.org/wiki/Kinetic_energy#Derivation_and_definition

This is the correct way to do it. To fix your equation, you have to consider that Work is an input of energy, so the $(v^2-u^2)$ shows a change in velocity, proportional to the change in work. Kinetic energy sets a reference point: $u=0 \implies E_k=0$ .
Title: Re: Deriving Ek
Post by: Ahmad on January 04, 2008, 10:07:42 am: To perhaps most accurately define work done you'd need to use line integrals, however think about this:

$\mbox{Work Done} = \int_{x=a}^{x=b} \mbox{F dx}$

$= \int_{x=a}^{x=b} \mbox{ma dx}$

$= \int_{v=a}^{v=v_0} \mbox{mv dv}$ (Since a = dv/dt = dv/dx * dx/dt = dv/dx * v)

$= \frac{1}{2} mv_0^2$ (We don't worry about v=a, since we want total KE, we want the work done from rest!).
Title: Re: Deriving Ek
Post by: Mao on January 04, 2008, 01:03:08 pm: riiight, that make sense now! thanks guys...
=D

and i have no idea how line integral works.... someday... ;D