I am not understanding this differential relationship

1. Aug 20, 2013

mindheavy

I'm studying engineering dynamics. The first chapter is discussing the velocity and acceleration equations; v = ds/dt and a = dv/dt. It then goes on to show a third equation that is stated as "a ds = v dv". They say they derived this equation by combining the two previous and 'eliminating dt'. I am just not seeing how they arrived at this, what are the intermediate steps? I also am not understanding the reason for just 'eliminating' dt. Could anyone develop this or help me along my way of understanding how this third equation was reached, I feel very uncomfortable just memorizing it without understanding where it came from...

2. Aug 20, 2013

saminator910

Pretty much solve as you would any other equation, the like term is dt, so it can be eliminated.

get both in terms of dt = something, then set equal to eliminate the term.

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

3. Aug 20, 2013

HallsofIvy

You can "eliminate the dt" by using the chain rule, a= dv/dt= (dv/ds)(ds/dt)= (dv/ds)v

From a/v= dv/ds, we get ads= vdv or, equivalently, ds/v= dv/a

4. Aug 20, 2013