I am not understanding this differential relationship

[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...

 

[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.

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

 

[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

 

[mindheavy]

Thanks for the reply saminator910 & HallsofIvy, very helpful!