Integration - Projectile Motion w/ Air Resistance

[crastinus]

I"ve seen several texts that say something like this:

Our equation of motion along x is
dv_x/dt = -g(v_x/v_t)

Integrating this, we obtain equation 178 at this link (pretty near the top; sorry, but I can't figure out how to put the equation here). It's from FitzPatrick's online notes for classical mechanics.

I don't see how he gets to 178.

 

[Doc Al]

Realize that $\frac{dv}{dt} = av$ can be rewritten as $\frac{dv}{v} = adt$.

 

[crastinus]

I'd learned that we shouldn't think of dx/dt as a quotient at all, and so we can't multiply both sides by dt.

I have seen enough to know that that's wrong, but I don't really understand why. Is this something to do with differentials?

Thanks for the help!

 

[Dr Transport]

That's how you solve differential equations...

 

[haushofer]

I'd learned that we shouldn't think of dx/dt as a quotient at all, and so we can't multiply both sides by dt.

I have seen enough to know that that's wrong, but I don't really understand why. Is this something to do with differentials?

Thanks for the help!

This is called separation of variables and its justification comes from the chain rule. It shows that in some cases you can treat a derivative as 'merely' a fraction of differentials.

 

[PeroK]

I'd learned that we shouldn't think of dx/dt as a quotient at all, and so we can't multiply both sides by dt.

I have seen enough to know that that's wrong, but I don't really understand why. Is this something to do with differentials?

Thanks for the help!

Try:

http://tutorial.math.lamar.edu/Classes/CalcI/Differentials.aspx