Equation of Motion for a Projectile Under Quadratic Air Resistance

[learningphysics]

at home... not here at school. Sorry. Why?

Just wanted you to read the substitution method for solving integrals... Have a look at this example online:

http://www.sosmath.com/calculus/integration/substitution/Example1/Example1.html

Let me know if it makes sense.

 

[Oblio]

Why is du divided by 2?

 

[learningphysics]

Ok, we went from tf-ti = $$\int$$ $$\frac{-mdv}{mg(1+(c/mg)v^2}$$

Bring out the constants $$\frac{-m}{mg}$$


=$$\frac{-m}{mg}$$ $$\int$$$$\frac{dv}{1+(c/mg)v^2}$$

the numerator and denominator are factored out above...

Yeah, let's cancel out the m's... leaving:

$$\frac{-1}{g}$$ $$\int$$$$\frac{dv}{1+(c/mg)v^2}$$

The idea is now I want this integral to look like:

$$\int\frac{dx}{1+x^2}$$

for that we need to substitute a new variable instead of v...

If I let u = $$\sqrt{c/mg}*v$$, then the denominator is 1+(c/mg)v^2 = 1+u^2... does this part make sense?

 

[learningphysics]

Why is du divided by 2?

du = 2xdx

du/2 = xdx

so they substitute in du/2 instead of xdx.

 

[Oblio]

Yeah, let's cancel out the m's... leaving:

$$\frac{-1}{g}$$ $$\int$$$$\frac{dv}{1+(c/mg)v^2}$$

The idea is now I want this integral to look like:

$$\int\frac{dx}{1+x^2}$$

for that we need to substitute a new variable instead of v...

If I let u = $$\sqrt{c/mg}*v$$, then the denominator is 1+(c/mg)v^2 = 1+u^2... does this part make sense?

u = $$\sqrt{c/mg}*v$$
= u^2 = (c/mg)v^2

Yep. I get it!

 

[Oblio]

I just changed it, you may have already started replying to my dumb mistake before the edit. :P

 

[Oblio]

du = 2xdx

du/2 = xdx

so they substitute in du/2 instead of xdx.

right. duh.
I'm going to stab my brain with a q-tip.

 

[Oblio]

Shouldn't the du integrate into a u?

 

[learningphysics]

Shouldn't the du integrate into a u?

write out the integral after substituting in u...

 

[learningphysics]

right. duh.
I'm going to stab my brain with a q-tip.

lol. don't be so hard on yourself.

 

[Oblio]

so you'd have:

u^76*u*1/152

=(u^76*x^2+5) / 152

 

[learningphysics]

so you'd have:

u^76*u*1/152

=(u^76*x^2+5) / 152

no... du doesn't integrate to u...

When you have an integral:

let's say:

$$\int5xdx$$

What you are taking the anti-derivative of is 5x... so the answer would be (5/2)x^2.

same way:

$$\int u^{75}(du/2) = \int \frac{u^{75}}{2} du$$

You take the anti-derivative of u^75/2 to get your answer.

 

[Oblio]

Ok, got it. It's more of a notation.

Yeah I follow your online example now.

 

[Oblio]

Yeah, let's cancel out the m's... leaving:

$$\frac{-1}{g}$$ $$\int$$$$\frac{dv}{1+(c/mg)v^2}$$

The idea is now I want this integral to look like:

$$\int\frac{dx}{1+x^2}$$

for that we need to substitute a new variable instead of v...

If I let u = $$\sqrt{c/mg}*v$$, then the denominator is 1+(c/mg)v^2 = 1+u^2... does this part make sense?

So, if u = $$\sqrt{c/mg}*v$$

How do we still end up with $$\sqrt{c/mg}$$ outside the integral as well?

 

[learningphysics]

So, if u = $$\sqrt{c/mg}*v$$

How do we still end up with $$\sqrt{c/mg}$$ outside the integral as well?

we not only need to substitute u... we also need to take care of the dv, and get du instead...

so:

du = $$\sqrt{c/mg}*dv$$

dv = $$\frac{du}{\sqrt{c/mg}}$$

So substitute that into the integral for dv...

what do you get for the integral now?

 

[learningphysics]

Ok, got it. It's more of a notation.

Yeah, for the most part...

Yeah I follow your online example now.

cool.

 

[Oblio]

we not only need to substitute u... we also need to take care of the dv, and get du instead...

so:

du = $$\sqrt{c/mg}*dv$$

dv = $$\frac{du}{\sqrt{c/mg}}$$

So substitute that into the integral for dv...

what do you get for the integral now?

$$\frac{-m}{mg}$$ $$\int$$ $$\frac{\frac{du}{sqrt[c/mg]}}{1 + c/mg* v^2}$$

 

[learningphysics]

$$\frac{-m}{mg}$$ $$\int$$ $$\frac{\frac{du}{sqrt[c/mg]}}{1 + c/mg* v^2}$$

yes, but you also need the u substitution...

 

[Oblio]

You mean du?

 

[learningphysics]

You mean du?

you've taken care of the du... but you still have v in the integral... you want u... we had it before

1+u^2

 

[Oblio]

$$\frac{-m}{mg}$$ $$\int$$ $$\frac{\frac{du}{sqrt[c/mg]}}{1 + c/mg* v^2}$$

right...

$$\frac{-m}{mg}$$ $$\int$$ $$\frac{\frac{du}{sqrt[c/mg]}}{1+ u^2}$$

 

[learningphysics]

right...

$$\frac{-m}{mg}$$ $$\int$$ $$\frac{\frac{du}{sqrt[c/mg]}}{1+ u^2}$$

exactly... now take all the constants outside the integral... and we get:

$$-\sqrt{\frac{m}{gc}}\int\frac{du}{1+u^2}$$

now we can apply arctan...

 

[Oblio]

Do you treat the integral sign almost as an equality when moving things in and out?

 

[learningphysics]

Do you treat the integral sign almost as an equality when moving things in and out?

not like an equal sign... more like a parentheses...

ie: 4x^2y+8xy = 4(x^2y + 2xy) = 4xy(x+2)

it's the same way I'm moving out the constants...

 

[Oblio]

Ok because I am having trouble matching what you got taking out the last constants

 

[Oblio]

what step did you take?

 

[learningphysics]

Ok because I am having trouble matching what you got taking out the last constants

show me what you get... I might have made a mistake.

 

[Oblio]

One more question first, how do we treat it like a parantheses, when you don't really have anything outside the 'bracket' to multiply or divide, in the simplification process.

I tried lots of things, but logically I'm not sure the proper logic behind taking out those constants when there's no 'other side' to multiply/divide etc...
know what i mean?

 

[learningphysics]

One more question first, how do we treat it like a parantheses, when you don't really have anything outside the 'bracket' to multiply or divide, in the simplification process.

I tried lots of things, but logically I'm not sure the proper logic behind taking out those constants when there's no 'other side' to multiply/divide etc...
know what i mean?

I'm not sure I understand... for example:

$$\int5xdx = 5\int xdx$$

For any integral $$\int A*f(x)dx = A\int f(x)dx$$ where A is a constant.

we're just factoring it out... so you don't need an other side...

never mind about taking the constants out...

suppose I had it like this:

$$\int\frac{-m}{mg}*\frac{\frac{du}{sqrt[c/mg]}}{1+ u^2}$$

now, I just want to clean this up a little... simplify it... don't take anything outsdie the integral.

 

[Oblio]

You mean for me to to it?

I know and understand your first rule there, but does that apply to a multi layer division?

 

[learningphysics]

You mean for me to to it?

Yeah.

I know and understand your first rule there, but does that apply to a multi layer division?

Which rule?

 

[Oblio]

$$\int5xdx = 5\int xdx$$

For any integral $$\int A*f(x)dx = A\int f(x)dx$$ where A is a constant.

QUOTE]

 

[learningphysics]

$$\int5xdx = 5\int xdx$$

For any integral $$\int A*f(x)dx = A\int f(x)dx$$ where A is a constant.

QUOTE]

don't worry about taking anything out of the integral... just simplify it inside the integral.

 

[Oblio]

Yeah, that's what I'm aiming for, but when I simplify equations; mentally I do the 'what you do to one side, do to the other' etc. With nothing outside of the integral, how do we bring out sqrt[c/mg] ?

 

[learningphysics]

Yeah, that's what I'm aiming for, but when I simplify equations; mentally I do the 'what you do to one side, do to the other' etc. With nothing outside of the integral, how do we bring out sqrt[c/mg] ?

don't bring it out:

$$\int\frac{-m}{mg}*\frac{\frac{du}{sqrt[c/mg]}}{1+ u^2}$$

this equals

$$\int\frac{-m}{mg}*\frac{1}{sqrt[c/mg]}*\frac{du}{1+ u^2}$$

now, the first thing you can do is cancel the m's in the numerator and denominator... what else can you do to simplify...