# Free fall with air drag force

1. Nov 6, 2004

### UrbanXrisis

free fall with drag force

If the equation F=-bv^2 describes the drag force of an object...then the differential equation for the object's motion would be:

dv/dt= -g+bv/m

or is it...

dv/dt= g-bv/m

After solving the equation, should I get...
V=mg/b[1-e^(-bt/m)]

Also, does this look like the position v time graph of the object?

Last edited: Nov 6, 2004
2. Nov 6, 2004

### Pyrrhus

Ok i'll do the analysis.

Forces acting on a body falling.
Down positive (Using resistive force proportional to the speed)

We got a first order DE

$$mg - bv = m \frac{dv}{dt}$$

To not get into much detail, this type of DE

$$\frac{dy}{dt} = ay - b$$

Has the following solution

$$y = \frac{b}{a} + ce^{at}$$

For an initial value, to find C.

$$y = \frac{b}{a} + [y_{o} - \frac{b}{a}]e^{at}$$

Thus for our case the solution is

$$v = \frac{mg}{b} + [v_{o} - \frac{mg}{b}]e^{-\frac{bt}{m}}$$

If we arrange the terms and $v_{o} = 0$

$$v = \frac{mg}{b} - \frac{mg}{b}e^{-\frac{bt}{m}}$$

$$v = \frac{mg}{b}(1 - e^{-\frac{bt}{m}})$$

Last edited: Nov 6, 2004
3. Nov 6, 2004

### UrbanXrisis

instead of F=-bv^2, what if F=-2v? The question tells me to ignore the effects of gravity in this problem. Would the DE then be dv/dt= 2v/m?

Since f=bv...and b=-2

dv/dt= g-bv/m
dv/dt= g-(-2)v/m
the question asks to ignore gravity...
dv/dt= -(-2)v/m
dv/dt= 2v/m

4. Nov 6, 2004

### Pyrrhus

If we ignore gravity, then indeed our Newtonian analysis will be

The Object is falling, and the air drag is in the opposite direction
Down positive.

$$m\frac{dv}{dt} = -bv$$

$$\frac{dv}{dt} = \frac{-bv}{m}$$

Solving this:

$$\frac{dv}{dt} = \frac{-bv}{m}$$

$$\frac{dv}{v} = \frac{-b}{m}dt$$

$$\int_{v_{o}}^{v} \frac{dv}{v} = \int_{0}^{t} \frac{-b}{m}dt$$

$$\ln |v|]_{v_{o}}^{v} = \frac{-b}{m}t]_{0}^{t}$$

For $v_{o} = 0$

$$ln |v| = \frac{-b}{m}t$$

$$v = e^{\frac{-b}{m}t}$$

Last edited: Nov 6, 2004
5. Nov 6, 2004

### UrbanXrisis

Does this look correct?

6. Nov 6, 2004

### UrbanXrisis

However, the retarding force is not F=bv, it is F=-2v as stated above. Correct me if I’m wrong, but isn’t $$\frac{dv}{dt} = \frac{-bv}{m}$$ for when F=bv? Should it be $$\frac{dv}{dt} = \frac{2v}{m}$$ ?

7. Nov 6, 2004

### Pyrrhus

Sure, just substitute b with 2, and for the graphic, It looks rather odd to me, but maybe someone else can find it the physical meaning.

Last edited: Nov 6, 2004
8. Nov 6, 2004

### UrbanXrisis

Last edited: Nov 6, 2004
9. Nov 6, 2004

### Pyrrhus

the graphic looks better now, and i added the extra solve steps for neglecting gravity.

10. Feb 10, 2010