Integrate Equation: Find Position with Initial Drag Force & Velocity

[Ashley1nOnly]

Homework Statement

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

Homework Equations

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

 

[eys_physics]

Homework Statement

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

Homework Equations

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

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

 

[Ashley1nOnly]

Ln(x)

 

[cnh1995]

dv/(-g-kv)=dt

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

 

[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

 

[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

 

[Ashley1nOnly]

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

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

 

[cnh1995]

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

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

 

[eys_physics]

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

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

 

[Buffu]

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

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

 

[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

 

[Mark44]

Homework Statement

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

First off, this isn't an equation -- an equation always has '=' somewhere in it.
Second, are V and v different quantities?

Homework Equations

The Attempt at a Solution

U substitution won't work

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.

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 it appears that V and v both represent velocity, a variable. Writing both V and v in the same expression is very confusing.

Now I want to integrate the left side which I don't know how to do.

Write the equation as $\int \frac {v~dv}{g + kv} = -\int dx$. On the left side, use polynomial long division.

 

[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

 

[cnh1995]

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

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.