Silly question, derivation of energy

[tuoni]

Here's a silly question I ended up thinking about. Maybe I learned it in college, but it's long forgotten by now ^~^; It might be quite a silly question, something I shouldn't have forgotten so easily, but alas, here I am, I've gotten so accustomed to simply using the algorithms without much thought. What a bad thing to happen T_T

Where did the "divided by 2" go?

${\color{red} a := \frac{m}{s^{2}} := \frac{m^{2}}{s^{2}m} := \frac{ \left( \frac{m^{2}}{s^{2}} \right) }{m} := \frac{ \left( \frac{m}{s} \right) \left( \frac{m}{s} \right) }{m}} \;\; \Rightarrow \;\; a = \frac{v^{2}}{s}$

$E = \frac{mv^{2}}{2}$

$E = Fs = mas = m \frac{v^{2}}{s} s = \frac{mv^{2}s}{s} = mv^{2}$

a := acceleration
E := energy
F := force
m := mass
m := metre
s := space
s := second
v := velocity
​

Is my mind deteriorating, or why am I unable to see what is wrong?

 

[Helios]

You're mixing up ( red ) s as seconds with ( black) s as displacement. See definition of Joule ( energy ) and Newton ( force ). In fact, some of your definitions are quite reckless. Get rid of m = metre. Prefer s = displacement. Get rid of s = second. Prefer t = time.

 

[rock.freak667]

In your finding an equation for accleration, you used dimensional analysis. Using that doesn't give you the value of constants, so it not safe to assume that the constant is always 1.

Your equation for acceleration should have been $a=\frac{kv^2}{s}$

But I see that you were deriving the formula for kinetic energy.
There are two ways to do it, with a constant force or variable force.

For constant force.

E=Fs=mas

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

If you wanted to use calculus (for a variable force)

$$E= \int F ds= \int (ma) ds$$

and $a=\frac{dv}{dt}$ (So use the chain rule to get 'a' in terms of d(something)/ds)

 

[Doc Al]

${\color{red} a := \frac{m}{s^{2}} := \frac{m^{2}}{s^{2}m} := \frac{ \left( \frac{m^{2}}{s^{2}} \right) }{m} := \frac{ \left( \frac{m}{s} \right) \left( \frac{m}{s} \right) }{m}} \;\; \Rightarrow \;\; a = \frac{v^{2}}{s}$

All this demonstrates is that v^2/s has the same units as acceleration. For an object uniformly accelerated from rest, $a = v^2/2s$ where v is the final speed. (Derive this from the definition of acceleration and average velocity.)

 

[tuoni]

I haven't really done much derivation in my life, so yes, never expected to get it right the first time. Demonstrating that the units are the same was probably not the best idea either. However, thank you for the help ^_^

I'm also using a algorithms/equations book from college, so I never realized to look elsewhere. A few things I haven't seen before. I took all the physics classes in college, either I have forgotten quite a bit, or there were some very important parts left out.

I am working on ballistic algorithms, so that is why I've suddenly began with derivation. It makes it a lot easier for users to understand the algorithms if they see how it's derived, rather than just in-your-face.