Question #300534

A differential equation for the velocity v of a falling mass m subjected to air resistance

proportional to the square of the instantaneous velocity is:

𝑚 (𝑑𝑣/𝑑𝑡) = mg-kv2

Where k >0 is a constant of proportionality. The positive direction is downward.

(a) Solve the equation subject to the initial condition v(0)=v0.

(b) Use the solution in part (a) to determine the limiting, or terminal, velocity of the mass.

(c) If the distance s, measured from the point where the mass was released above ground, is

related to velocity v by ds/dt=v(t), find an explicit expression for s(t) if s(0)= 0.


1
Expert's answer
2022-02-24T15:43:50-0500

𝑚(𝑑𝑣/𝑑𝑡)=mgkv2𝑚(𝑑𝑣/𝑑𝑡) = mg-kv^2

Put mg=ku02mg=ku_0^2 and u0=gTu_0=gT. Then

𝑚(𝑑𝑣/𝑑𝑡)=k(u02v2)𝑚 (𝑑𝑣/𝑑𝑡) = k(u_0^2-v^2)


1. Consider the case u0>v0u_0>v_0

dvu02v2=kmdt\frac{dv}{u_0^2-v^2}=\frac{k}{m}dt

v(0)v(t)dvu02v2=0tkmdt=ktm\int\limits_{v(0)}^{v(t)}\frac{dv}{u_0^2-v^2}=\int\limits_{0}^{t}\frac{k}{m}dt=\frac{kt}{m}

v(0)v(t)dvu02v2=v0v(t)(1u0v+1u0+v)dv2u0=\int\limits_{v(0)}^{v(t)}\frac{dv}{u_0^2-v^2}=\int\limits_{v_0}^{v(t)}\left(\frac{1}{u_0-v}+\frac{1}{u_0+v}\right)\frac{dv}{2u_0}=

=12u0logu0+v(t)u0v(t)12u0logu0+v0u0v0=\frac{1}{2u_0} \log\frac{u_0+v(t)}{u_0-v(t)}-\frac{1}{2u_0} \log\frac{u_0+v_0}{u_0-v_0}

logu0+v(t)u0v(t)=logu0+v0u0v0+2tku0m\log\frac{u_0+v(t)}{u_0-v(t)}=\log\frac{u_0+v_0}{u_0-v_0}+2t\frac{ku_0}{m}

But ku0m=gku0gm=gku0ku02=gu0=1T\frac{ku_0}{m}=\frac{gku_0}{gm}=\frac{gku_0}{ku_0^2}=\frac{g}{u_0}=\frac{1}{T}

u0+v(t)u0v(t)=u0+v0u0v0e2kv0tm=u0+v0u0v0e2tT=e2(tt0)T\frac{u_0+v(t)}{u_0-v(t)}=\frac{u_0+v_0}{u_0-v_0}e^{\frac{2kv_0t}{m}}=\frac{u_0+v_0}{u_0-v_0}e^{\frac{2t}{T}}=e^{\frac{2(t-t_0)}{T}} ,

where t0=12logu0+v0u0v0t_0=-\frac{1}{2}\log\frac{u_0+v_0}{u_0-v_0} is the time when v(t)=0v(t)=0. Then

2u0u0v(t)=1+u0+v(t)u0v(t)=1+e2(tt0)T\frac{2u_0}{u_0-v(t)}=1+\frac{u_0+v(t)}{u_0-v(t)}=1+e^{\frac{2(t-t_0)}{T}}

u0v(t)=2u01+e2(tt0)Tu_0-v(t)=\frac{2u_0}{1+e^{\frac{2(t-t_0)}{T}}}

v(t)=u02u01+e2(tt0)T=u0e2(tt0)T1e2(tt0)T+1=u0tanhtt0Tv(t)=u_0-\frac{2u_0}{1+e^{\frac{2(t-t_0)}{T}}}=u_0\frac{e^{\frac{2(t-t_0)}{T}}-1}{e^{\frac{2(t-t_0)}{T}}+1}=u_0\tanh\frac{t-t_0}{T}

where u0=mgku_0=\sqrt\frac{mg}{k}, T=u0g=mkgT=\frac{u_0}{g}=\sqrt\frac{m}{kg} and t0=12logu0+v0u0v0t_0=-\frac{1}{2}\log\frac{u_0+v_0}{u_0-v_0}.


(b) limt+v(t)=limt+u0tanhtt0T=u0=mgk\lim\limits_{t\to+\infty}v(t)=\lim\limits_{t\to+\infty}u_0\tanh\frac{t-t_0}{T}=u_0=\sqrt\frac{mg}{k}


(c) s(t)=s(0)+0tv(t)dt=0tu0tanhtt0Tdt=s(t)=s(0)+\int\limits_{0}^{t}v(t)dt=\int\limits_{0}^{t}u_0\tanh\frac{t-t_0}{T}dt=

=u0T0tdcoshtt0Tcoshtt0T=u0T(logcoshtt0Tlogcosht0T)=u_0T\int\limits_{0}^{t}\frac{d\cosh\frac{t-t_0}{T}}{\cosh\frac{t-t_0}{T}}=u_0T(\log\cosh\frac{t-t_0}{T}-\log\cosh\frac{t_0}{T})

=u0Tlogcoshtt0Tcosht0T=u_0T\log\frac{\cosh\frac{t-t_0}{T}}{\cosh\frac{t_0}{T}}


2. Consider the case u0<v0u_0<v_0

dvu02v2=kmdt\frac{dv}{u_0^2-v^2}=\frac{k}{m}dt

v(0)v(t)dvu02v2=0tkmdt=ktm\int\limits_{v(0)}^{v(t)}\frac{dv}{u_0^2-v^2}=\int\limits_{0}^{t}\frac{k}{m}dt=\frac{kt}{m}

v(0)v(t)dvu02v2=v0v(t)(1u0v+1u0+v)dv2u0=\int\limits_{v(0)}^{v(t)}\frac{dv}{u_0^2-v^2}=\int\limits_{v_0}^{v(t)}\left(\frac{1}{u_0-v}+\frac{1}{u_0+v}\right)\frac{dv}{2u_0}=

=12u0logv(t)+u0v(t)u012u0logv0+u0v0u0=\frac{1}{2u_0} \log\frac{v(t)+u_0}{v(t)-u_0}-\frac{1}{2u_0} \log\frac{v_0+u_0}{v_0-u_0}

logv(t)+u0v(t)u0=logv0+u0v0u0+2tku0m=logv0+u0v0u0+2tT\log\frac{v(t)+u_0}{v(t)-u_0}=\log\frac{v_0+u_0}{v_0-u_0}+2t\frac{ku_0}{m}=\log\frac{v_0+u_0}{v_0-u_0}+\frac{2t}{T}

v(t)+u0v(t)u0=v0+u0v0u0e2tT=e2(tt0)T\frac{v(t)+u_0}{v(t)-u_0}=\frac{v_0+u_0}{v_0-u_0}e^{\frac{2t}{T}}=e^{\frac{2(t-t_0)}{T}} ,

where t0=12logv0+u0v0u0t_0=-\frac{1}{2}\log\frac{v_0+u_0}{v_0-u_0} is the time when v(t)=v(t)=\infty. Then

2u0v(t)u0=v(t)+u0v(t)u01=e2(tt0)T1\frac{2u_0}{v(t)-u_0}=\frac{v(t)+u_0}{v(t)-u_0}-1=e^{\frac{2(t-t_0)}{T}}-1

v(t)u0=2u0e2(tt0)T1v(t)-u_0=\frac{2u_0}{e^{\frac{2(t-t_0)}{T}}-1}

v(t)=u0+2u0e2(tt0)T1=u0e2(tt0)T+1e2(tt0)T1=u0cothtt0Tv(t)=u_0+\frac{2u_0}{e^{\frac{2(t-t_0)}{T}}-1}=u_0\frac{e^{\frac{2(t-t_0)}{T}}+1}{e^{\frac{2(t-t_0)}{T}}-1}=u_0\coth\frac{t-t_0}{T}

where u0=mgku_0=\sqrt\frac{mg}{k}, T=u0g=mkgT=\frac{u_0}{g}=\sqrt\frac{m}{kg} and t0=12logv0+u0v0u0t_0=-\frac{1}{2}\log\frac{v_0+u_0}{v_0-u_0}.


3. Consider the case v0=u0v_0=u_0. It is evident that in this case v(t)=u0v(t)=u_0 is the unique solution.


(b) limt+v(t)=limt+u0tanhtt0T=u0\lim\limits_{t\to+\infty}v(t)=\lim\limits_{t\to+\infty}u_0\tanh\frac{t-t_0}{T}=u_0 , if u0>v0u_0>v_0 ,

limt+v(t)=limt+u0cothtt0T=u0\lim\limits_{t\to+\infty}v(t)=\lim\limits_{t\to+\infty}u_0\coth\frac{t-t_0}{T}=u_0 , if u0<v0u_0<v_0 ,

limt+v(t)=limt+u0=u0\lim\limits_{t\to+\infty}v(t)=\lim\limits_{t\to+\infty}u_0=u_0 , if u0=v0u_0=v_0 .

Therefore, in any case limt+v(t)=u0=mgk\lim\limits_{t\to+\infty}v(t)=u_0=\sqrt\frac{mg}{k}.


(c) s(t)=s(0)+0tv(t)dt=0tu0tanhtt0Tdt=s(t)=s(0)+\int\limits_{0}^{t}v(t)dt=\int\limits_{0}^{t}u_0\tanh\frac{t-t_0}{T}dt=

=u0T0tdcoshtt0Tcoshtt0T=u0T(logcoshtt0Tlogcosht0T)=u_0T\int\limits_{0}^{t}\frac{d\cosh\frac{t-t_0}{T}}{\cosh\frac{t-t_0}{T}}=u_0T(\log\cosh\frac{t-t_0}{T}-\log\cosh\frac{t_0}{T})

=u0Tlogcoshtt0Tcosht0T=u_0T\log\frac{\cosh\frac{t-t_0}{T}}{\cosh\frac{t_0}{T}} , if u0>v0u_0>v_0

If u0<v0u_0<v_0 then s(t)=s(0)+0tv(t)dt=0tu0cothtt0Tdt=s(t)=s(0)+\int\limits_{0}^{t}v(t)dt=\int\limits_{0}^{t}u_0\coth\frac{t-t_0}{T}dt=

=u0T0tdsinhtt0Tsinhtt0T=u0T(logsinhtt0Tlogsinht0T)=u_0T\int\limits_{0}^{t}\frac{d\sinh\frac{t-t_0}{T}}{\sinh\frac{t-t_0}{T}}=u_0T(\log\sinh\frac{t-t_0}{T}-\log\sinh\frac{t_0}{T})

=u0Tlogsinhtt0Tsinht0T=u_0T\log\frac{\sinh\frac{t-t_0}{T}}{\sinh\frac{t_0}{T}}

If u0=v0u_0=v_0 then s(t)=s(0)+0tv(t)dt=0tu0dt=u0ts(t)=s(0)+\int\limits_{0}^{t}v(t)dt=\int\limits_{0}^{t}u_0dt=u_0t

Answer.

if u0>v0u_0>v_0 then

(a) v(t)=u0tanhtt0Tv(t)=u_0\tanh\frac{t-t_0}{T}, where u0=mgku_0=\sqrt\frac{mg}{k}, T=u0g=mkgT=\frac{u_0}{g}=\sqrt\frac{m}{kg} and t0=12logu0+v0u0v0t_0=-\frac{1}{2}\log\frac{u_0+v_0}{u_0-v_0}

(b) limt+v(t)=mgk\lim\limits_{t\to+\infty}v(t)=\sqrt\frac{mg}{k}

(c) s(t)=u0Tlogcoshtt0Tcosht0Ts(t)=u_0T\log\frac{\cosh\frac{t-t_0}{T}}{\cosh\frac{t_0}{T}}

if u0<v0u_0<v_0 then

(a) v(t)=u0cothtt0Tv(t)=u_0\coth\frac{t-t_0}{T}, where u0=mgku_0=\sqrt\frac{mg}{k}, T=u0g=mkgT=\frac{u_0}{g}=\sqrt\frac{m}{kg} and t0=12logv0+u0v0u0t_0=-\frac{1}{2}\log\frac{v_0+u_0}{v_0-u_0}

(b) limt+v(t)=mgk\lim\limits_{t\to+\infty}v(t)=\sqrt\frac{mg}{k}

(c) s(t)=u0Tlogsinhtt0Tsinht0Ts(t)=u_0T\log\frac{\sinh\frac{t-t_0}{T}}{\sinh\frac{t_0}{T}}

if u0=v0u_0=v_0 then

(a) v(t)=v0=u0v(t)=v_0=u_0

(b) limt+v(t)=mgk\lim\limits_{t\to+\infty}v(t)=\sqrt\frac{mg}{k} ,

(c) s(t)=u0ts(t)=u_0t


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