# Integrate equation

1. Dec 4, 2016

### Ashley1nOnly

1. The problem statement, all variables and given/known data
I'm trying to integrate this equation so that I can find my position using my initial drag force and initial velocity.

V/(-g-kv) dv
2. Relevant equations

3. The attempt at a solution
U substitution won't work
Integration by parts won't work
Partial fractions won't work.
It doesn't match any trigonometric equations

I'm not sure how to integrate it.
Basically a potato is being shot up in the air with air resistance and gravity affecting it.
This is what I started with
F=ma
F=-mg-kmv
M(dv/dt)= -mg-kmv
dv/dt=-g-kv
dv/(-g-kv)=dt
well v=dx/dt so dt=dx/v

dv/(-g-kv)=dx/v
Separation of variable from D.E
Vdv/(-g-kv)=dx
Now I want to integrate the left side which I don't know how to do.

Thanks for any help

Last edited: Dec 4, 2016
2. Dec 4, 2016

### eys_physics

Do you know have to compute the indefinite integral
$$\int \frac{dx}{x}\quad ?$$

3. Dec 4, 2016

Ln(x)

4. Dec 4, 2016

### cnh1995

This is the equation you should integrate.
Are you asked to find the displacement as a function of time?

5. Dec 4, 2016

### Ashley1nOnly

If the
V dv
-------------
(-g-kv)

If the v wasn't at the stop I could integrate it but I don't know what to do with the V at the top

6. Dec 4, 2016

### Ashley1nOnly

I'm not given time which is why I had to use the substitution dt=dx/v. Because I am given initial velocity and k

7. Dec 4, 2016

### Ashley1nOnly

I did that integral first then realized I didn't have time so I had to take a different route

8. Dec 4, 2016

### cnh1995

Ok.
So,
v*dv/(g+kv)=-dx
Hint: Isolate the v in the denominator (get rid of the k attached to it).

9. Dec 4, 2016

### eys_physics

Take first out the minus sign from the denominator. Do then the substitution $x=g+kv$.

10. Dec 4, 2016

### Buffu

just a suggestion :- try binomial expansion on the denominator.

11. Dec 4, 2016

### Ashley1nOnly

So I did some rearrangements and got

dv
--------- =- dx
g/v +k

Then integrating both sides I get
Ln((g/v)+k)+c=-x

Assuming c is the initial velocity

12. Dec 4, 2016

### Staff: Mentor

First off, this isn't an equation -- an equation always has '=' somewhere in it.
Second, are V and v different quantities?
Substitution will work if V and v are different. If they are different, I'm assuming that V is a constant, and v is the variable velocity. If that's the case, substitution will work.
Now it appears that V and v both represent velocity, a variable. Writing both V and v in the same expression is very confusing.
Write the equation as $\int \frac {v~dv}{g + kv} = -\int dx$. On the left side, use polynomial long division.

13. Dec 4, 2016

### Ashley1nOnly

Sorry when I type in the first letter it always wants to capitalize it. Well I shall search for understanding on polynomial long division. Thank you

14. Dec 4, 2016

### cnh1995

That's not correct.
Try the methods in #9 and #12. If it's not too much, you can try this too.. Take the k out from the denominator and you'll have the equation in the form y*dy/(a+y). Just a simple rearrangement in the numerator and it will be integrable.