Hi,

So I was trying to develop a more in-depth understanding of the constant acceleration equations, especially

I found a video showing the derivation of this formula: [youtube]JnFykw00HvE[/youtube]

His logic goes like this.

1: Acceleration is the derivative of velocity w.r.t. time.

2, Chain rule:

3, as ds/dt = v we have:

4,

5, integrating:

6, at t = 0, s = 0 and v = u (initial velocity) so

All well and good, right?

Except that in stage 4 he multiplies by "ds". I know that this is common practice amongst all you calculus pros, but strictly speaking "dv/ds" is a limit, not a fraction, right? So you can't treat ds like it was a denominator.

To better understand what is actually going on here, I decided to try and write an analogous argument using functional notation. And that's when I got stuck.

What'cha reckon? Here are my current thoughts:

a and u are constants

t is the independent variable.

v is a function, f, of t, so that

s is a function, g, of t, so that

v can also be written as a function, h, of s, so that

... so far so good?

We then have:

In stage 3., we had:

In functional notation this would be:

So I guess that h'(s) must equal a/v, but I can't explain that at all.

Anyone got any ideas?

Thanks!

Theo

So I was trying to develop a more in-depth understanding of the constant acceleration equations, especially

**v^2 = u^2+2as**.I found a video showing the derivation of this formula: [youtube]JnFykw00HvE[/youtube]

His logic goes like this.

1: Acceleration is the derivative of velocity w.r.t. time.

**a = dv/dt**2, Chain rule:

**a = dv/ds * ds/dt**

3, as ds/dt = v we have:

**a = v * dv/ds**4,

*Times by ds***a ds = v * dv**5, integrating:

**integral of a w.r.t. s = integral of v w.r.t. v**6, at t = 0, s = 0 and v = u (initial velocity) so

**as - a0 = 1/2 * v^2 - 1/2 * u^2****2as = v^2 - u^2****v^2 = u^2 + 2as**All well and good, right?

Except that in stage 4 he multiplies by "ds". I know that this is common practice amongst all you calculus pros, but strictly speaking "dv/ds" is a limit, not a fraction, right? So you can't treat ds like it was a denominator.

To better understand what is actually going on here, I decided to try and write an analogous argument using functional notation. And that's when I got stuck.

What'cha reckon? Here are my current thoughts:

a and u are constants

*(u is initial velocity - I think that might be British notation?)*.t is the independent variable.

v is a function, f, of t, so that

**f(t) = u + at**s is a function, g, of t, so that

**g(t) = ut + 1/2 * a * t^2**v can also be written as a function, h, of s, so that

**h(s) = Dt s***(Differentiate s wrt. t)*... so far so good?

We then have:

**fÂ´(t) = a**and**gÂ´(t) = u + at**but I can't figure out what**hÂ´(s) = ?**In stage 3., we had:

**a = v * dv/ds**In functional notation this would be:

**a = v * h'(s)**... right?So I guess that h'(s) must equal a/v, but I can't explain that at all.

Anyone got any ideas?

Thanks!

Theo

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