Calculating the Solution for a Falling Object with Air Resistance

[Opressor]

Homework Statement

A body falling under the action of the drag force of the air -βvⁿ
n ∈ ℤ
m= mass
v= velocity
g= 9,8m/s²
β= coefficient of air resistance

Homework Equations

m(dv/dt)= mg-βvⁿ

The Attempt at a Solution

I can not resolve to any integer (n)

 

[Opressor]

$$m\frac {dv} {dt} =mg- β v^n$$

 

[Delta2]

You can use separation of variables to solve the case for n=2 (maybe it can be applied to higher n as well).
$\frac{dv}{dt}=g(1-\frac{\beta}{mg}v^2)\Rightarrow \frac{dv}{g(1-\frac{\beta}{mg}v^2)}=dt$ and now all you have to do is integrate both sides.
You can take a look at https://en.wikipedia.org/wiki/Terminal_velocity#Derivation_for_terminal_velocity if you want more details (click show on that derivation link in the Wikipedia page).

 

[Opressor]

You can use separation of variables to solve the case for n=2 (maybe it can be applied to higher n as well).
$\frac{dv}{dt}=g(1-\frac{\beta}{mg}v^2)\Rightarrow \frac{dv}{g(1-\frac{\beta}{mg}v^2)}=dt$ and now all you have to do is integrate both sides.
You can take a look at https://en.wikipedia.org/wiki/Terminal_velocity#Derivation_for_terminal_velocity if you want more details (click show on that derivation link in the Wikipedia page).

thank you

 

[Opressor]

Does it treat this equation as Bernoulli's equation?

$$\frac {dy} {dx}\ +P(x)=Q(x)y^n$$

 

[Delta2]

Bernoulli's differential equation is slightly different, it has a y-term it is $\frac{dy}{dx}+P(x)y=Q(x)y^n$.

Are you sure that n is allowed to take values higher than 2? Because the drag force cases I know are either n=1 or n=2, never heard of a drag force $-bv^3$.

In case that $n\geq 3$ it seems that the solution involves calculation of the following integral

$$\int \frac{dv}{1-av^n}$$

which doesn't have a closed form, so I believe (I am not completely sure) that for $n\geq 3$ the solution doesn't have a closed form.

 

[Ray Vickson]

Bernoulli's differential equation is slightly different, it has a y-term it is $\frac{dy}{dx}+P(x)y=Q(x)y^n$.

Are you sure that n is allowed to take values higher than 2? Because the drag force cases I know are either n=1 or n=2, never heard of a drag force $-bv^3$.

In case that $n\geq 3$ it seems that the solution involves calculation of the following integral

$$\int \frac{dv}{1-av^n}$$

which doesn't have a closed form, so I believe (I am not completely sure) that for $n\geq 3$ the solution doesn't have a closed form.

If $J_n = \int 1/(1-a v^n) \, dv,$ then
Maple gets J34 = -(1/3)*ln(v-(1/a)^(1/3))/(a*(1/a)^(2/3))+(1/6)*ln(v^2+v*(1/a)^(1/3)+(1/a)^(2/3))/(a*(1/a)^(2/3))+(1/3)*3^(1/2)*arctan((1/3)*3^(1/2)*(2*v/(1/a)^(1/3)+1))/(a*(1/a)^(2/3))
and
J4 = (1/2)*(1/a)^(1/4)*arctan(v/(1/a)^(1/4))+(1/4)*(1/a)^(1/4)*ln((v+(1/a)^(1/4))/(v-(1/a)^(1/4)))

Furthermore, these check out when differentiating them wrt v.

Both of these are closed forms.

What is even more surprising to me is that Maple gets a (lengthy and horrible) closed-form for J5.

 

[Delta2]

Well ok, to be honest I check it with wolfram and it showed as result the incomplete beta function of some sort.

Even if $J_n$ has closed form, we then have to solve the equation $J_n(v)=t+C\Rightarrow v=…$ which I believe would not be possible, so the solution will not be in explicit form.

 

[vela]

In case that $n\geq 3$ it seems that the solution involves calculation of the following integral

$$\int \frac{dv}{1-av^n}$$

which doesn't have a closed form, so I believe (I am not completely sure) that for $n\geq 3$ the solution doesn't have a closed form.

You could expand the integrand using partial fractions and then integrate each term. It's probably not that bad of a calculation, actually, now that I think about it.