𝑚 ( 𝑑 𝑣 / 𝑑 𝑡 ) = m g − k v 2 𝑚(𝑑𝑣/𝑑𝑡) = mg-kv^2 m ( d v / d t ) = m g − k v 2
Put m g = k u 0 2 mg=ku_0^2 m g = k u 0 2 and u 0 = g T u_0=gT u 0 = g T . Then
𝑚 ( 𝑑 𝑣 / 𝑑 𝑡 ) = k ( u 0 2 − v 2 ) 𝑚 (𝑑𝑣/𝑑𝑡) = k(u_0^2-v^2) m ( d v / d t ) = k ( u 0 2 − v 2 )
1. Consider the case u 0 > v 0 u_0>v_0 u 0 > v 0
d v u 0 2 − v 2 = k m d t \frac{dv}{u_0^2-v^2}=\frac{k}{m}dt u 0 2 − v 2 d v = m k d t
∫ v ( 0 ) v ( t ) d v u 0 2 − v 2 = ∫ 0 t k m d t = k t m \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 ) u 0 2 − v 2 d v = 0 ∫ t m k d t = m k t
∫ v ( 0 ) v ( t ) d v u 0 2 − v 2 = ∫ v 0 v ( t ) ( 1 u 0 − v + 1 u 0 + v ) d v 2 u 0 = \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}= v ( 0 ) ∫ v ( t ) u 0 2 − v 2 d v = v 0 ∫ v ( t ) ( u 0 − v 1 + u 0 + v 1 ) 2 u 0 d v =
= 1 2 u 0 log u 0 + v ( t ) u 0 − v ( t ) − 1 2 u 0 log u 0 + v 0 u 0 − v 0 =\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} = 2 u 0 1 log u 0 − v ( t ) u 0 + v ( t ) − 2 u 0 1 log u 0 − v 0 u 0 + v 0
log u 0 + v ( t ) u 0 − v ( t ) = log u 0 + v 0 u 0 − v 0 + 2 t k u 0 m \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} log u 0 − v ( t ) u 0 + v ( t ) = log u 0 − v 0 u 0 + v 0 + 2 t m k u 0
But k u 0 m = g k u 0 g m = g k u 0 k u 0 2 = g u 0 = 1 T \frac{ku_0}{m}=\frac{gku_0}{gm}=\frac{gku_0}{ku_0^2}=\frac{g}{u_0}=\frac{1}{T} m k u 0 = g m g k u 0 = k u 0 2 g k u 0 = u 0 g = T 1
u 0 + v ( t ) u 0 − v ( t ) = u 0 + v 0 u 0 − v 0 e 2 k v 0 t m = u 0 + v 0 u 0 − v 0 e 2 t T = e 2 ( t − t 0 ) 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}} u 0 − v ( t ) u 0 + v ( t ) = u 0 − v 0 u 0 + v 0 e m 2 k v 0 t = u 0 − v 0 u 0 + v 0 e T 2 t = e T 2 ( t − t 0 ) ,
where t 0 = − 1 2 log u 0 + v 0 u 0 − v 0 t_0=-\frac{1}{2}\log\frac{u_0+v_0}{u_0-v_0} t 0 = − 2 1 log u 0 − v 0 u 0 + v 0 is the time when v ( t ) = 0 v(t)=0 v ( t ) = 0 . Then
2 u 0 u 0 − v ( t ) = 1 + u 0 + v ( t ) u 0 − v ( t ) = 1 + e 2 ( t − t 0 ) 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}} u 0 − v ( t ) 2 u 0 = 1 + u 0 − v ( t ) u 0 + v ( t ) = 1 + e T 2 ( t − t 0 )
u 0 − v ( t ) = 2 u 0 1 + e 2 ( t − t 0 ) T u_0-v(t)=\frac{2u_0}{1+e^{\frac{2(t-t_0)}{T}}} u 0 − v ( t ) = 1 + e T 2 ( t − t 0 ) 2 u 0
v ( t ) = u 0 − 2 u 0 1 + e 2 ( t − t 0 ) T = u 0 e 2 ( t − t 0 ) T − 1 e 2 ( t − t 0 ) T + 1 = u 0 tanh t − t 0 T v(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} v ( t ) = u 0 − 1 + e T 2 ( t − t 0 ) 2 u 0 = u 0 e T 2 ( t − t 0 ) + 1 e T 2 ( t − t 0 ) − 1 = u 0 tanh T t − t 0
where u 0 = m g k u_0=\sqrt\frac{mg}{k} u 0 = k m g , T = u 0 g = m k g T=\frac{u_0}{g}=\sqrt\frac{m}{kg} T = g u 0 = k g m and t 0 = − 1 2 log u 0 + v 0 u 0 − v 0 t_0=-\frac{1}{2}\log\frac{u_0+v_0}{u_0-v_0} t 0 = − 2 1 log u 0 − v 0 u 0 + v 0 .
(b) lim t → + ∞ v ( t ) = lim t → + ∞ u 0 tanh t − t 0 T = u 0 = m g k \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} t → + ∞ lim v ( t ) = t → + ∞ lim u 0 tanh T t − t 0 = u 0 = k m g
(c) s ( t ) = s ( 0 ) + ∫ 0 t v ( t ) d t = ∫ 0 t u 0 tanh t − t 0 T d t = s(t)=s(0)+\int\limits_{0}^{t}v(t)dt=\int\limits_{0}^{t}u_0\tanh\frac{t-t_0}{T}dt= s ( t ) = s ( 0 ) + 0 ∫ t v ( t ) d t = 0 ∫ t u 0 tanh T t − t 0 d t =
= u 0 T ∫ 0 t d cosh t − t 0 T cosh t − t 0 T = u 0 T ( log cosh t − t 0 T − log cosh t 0 T ) =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}) = u 0 T 0 ∫ t c o s h T t − t 0 d c o s h T t − t 0 = u 0 T ( log cosh T t − t 0 − log cosh T t 0 )
= u 0 T log cosh t − t 0 T cosh t 0 T =u_0T\log\frac{\cosh\frac{t-t_0}{T}}{\cosh\frac{t_0}{T}} = u 0 T log c o s h T t 0 c o s h T t − t 0
2. Consider the case u 0 < v 0 u_0<v_0 u 0 < v 0
d v u 0 2 − v 2 = k m d t \frac{dv}{u_0^2-v^2}=\frac{k}{m}dt u 0 2 − v 2 d v = m k d t
∫ v ( 0 ) v ( t ) d v u 0 2 − v 2 = ∫ 0 t k m d t = k t m \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 ) u 0 2 − v 2 d v = 0 ∫ t m k d t = m k t
∫ v ( 0 ) v ( t ) d v u 0 2 − v 2 = ∫ v 0 v ( t ) ( 1 u 0 − v + 1 u 0 + v ) d v 2 u 0 = \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}= v ( 0 ) ∫ v ( t ) u 0 2 − v 2 d v = v 0 ∫ v ( t ) ( u 0 − v 1 + u 0 + v 1 ) 2 u 0 d v =
= 1 2 u 0 log v ( t ) + u 0 v ( t ) − u 0 − 1 2 u 0 log v 0 + u 0 v 0 − u 0 =\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} = 2 u 0 1 log v ( t ) − u 0 v ( t ) + u 0 − 2 u 0 1 log v 0 − u 0 v 0 + u 0
log v ( t ) + u 0 v ( t ) − u 0 = log v 0 + u 0 v 0 − u 0 + 2 t k u 0 m = log v 0 + u 0 v 0 − u 0 + 2 t T \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} log v ( t ) − u 0 v ( t ) + u 0 = log v 0 − u 0 v 0 + u 0 + 2 t m k u 0 = log v 0 − u 0 v 0 + u 0 + T 2 t
v ( t ) + u 0 v ( t ) − u 0 = v 0 + u 0 v 0 − u 0 e 2 t T = e 2 ( t − t 0 ) 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}} v ( t ) − u 0 v ( t ) + u 0 = v 0 − u 0 v 0 + u 0 e T 2 t = e T 2 ( t − t 0 ) ,
where t 0 = − 1 2 log v 0 + u 0 v 0 − u 0 t_0=-\frac{1}{2}\log\frac{v_0+u_0}{v_0-u_0} t 0 = − 2 1 log v 0 − u 0 v 0 + u 0 is the time when v ( t ) = ∞ v(t)=\infty v ( t ) = ∞ . Then
2 u 0 v ( t ) − u 0 = v ( t ) + u 0 v ( t ) − u 0 − 1 = e 2 ( t − t 0 ) T − 1 \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 ) − u 0 2 u 0 = v ( t ) − u 0 v ( t ) + u 0 − 1 = e T 2 ( t − t 0 ) − 1
v ( t ) − u 0 = 2 u 0 e 2 ( t − t 0 ) T − 1 v(t)-u_0=\frac{2u_0}{e^{\frac{2(t-t_0)}{T}}-1} v ( t ) − u 0 = e T 2 ( t − t 0 ) − 1 2 u 0
v ( t ) = u 0 + 2 u 0 e 2 ( t − t 0 ) T − 1 = u 0 e 2 ( t − t 0 ) T + 1 e 2 ( t − t 0 ) T − 1 = u 0 coth t − t 0 T v(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} v ( t ) = u 0 + e T 2 ( t − t 0 ) − 1 2 u 0 = u 0 e T 2 ( t − t 0 ) − 1 e T 2 ( t − t 0 ) + 1 = u 0 coth T t − t 0
where u 0 = m g k u_0=\sqrt\frac{mg}{k} u 0 = k m g , T = u 0 g = m k g T=\frac{u_0}{g}=\sqrt\frac{m}{kg} T = g u 0 = k g m and t 0 = − 1 2 log v 0 + u 0 v 0 − u 0 t_0=-\frac{1}{2}\log\frac{v_0+u_0}{v_0-u_0} t 0 = − 2 1 log v 0 − u 0 v 0 + u 0 .
3. Consider the case v 0 = u 0 v_0=u_0 v 0 = u 0 . It is evident that in this case v ( t ) = u 0 v(t)=u_0 v ( t ) = u 0 is the unique solution.
(b) lim t → + ∞ v ( t ) = lim t → + ∞ u 0 tanh t − t 0 T = u 0 \lim\limits_{t\to+\infty}v(t)=\lim\limits_{t\to+\infty}u_0\tanh\frac{t-t_0}{T}=u_0 t → + ∞ lim v ( t ) = t → + ∞ lim u 0 tanh T t − t 0 = u 0 , if u 0 > v 0 u_0>v_0 u 0 > v 0 ,
lim t → + ∞ v ( t ) = lim t → + ∞ u 0 coth t − t 0 T = u 0 \lim\limits_{t\to+\infty}v(t)=\lim\limits_{t\to+\infty}u_0\coth\frac{t-t_0}{T}=u_0 t → + ∞ lim v ( t ) = t → + ∞ lim u 0 coth T t − t 0 = u 0 , if u 0 < v 0 u_0<v_0 u 0 < v 0 ,
lim t → + ∞ v ( t ) = lim t → + ∞ u 0 = u 0 \lim\limits_{t\to+\infty}v(t)=\lim\limits_{t\to+\infty}u_0=u_0 t → + ∞ lim v ( t ) = t → + ∞ lim u 0 = u 0 , if u 0 = v 0 u_0=v_0 u 0 = v 0 .
Therefore, in any case lim t → + ∞ v ( t ) = u 0 = m g k \lim\limits_{t\to+\infty}v(t)=u_0=\sqrt\frac{mg}{k} t → + ∞ lim v ( t ) = u 0 = k m g .
(c) s ( t ) = s ( 0 ) + ∫ 0 t v ( t ) d t = ∫ 0 t u 0 tanh t − t 0 T d t = s(t)=s(0)+\int\limits_{0}^{t}v(t)dt=\int\limits_{0}^{t}u_0\tanh\frac{t-t_0}{T}dt= s ( t ) = s ( 0 ) + 0 ∫ t v ( t ) d t = 0 ∫ t u 0 tanh T t − t 0 d t =
= u 0 T ∫ 0 t d cosh t − t 0 T cosh t − t 0 T = u 0 T ( log cosh t − t 0 T − log cosh t 0 T ) =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}) = u 0 T 0 ∫ t c o s h T t − t 0 d c o s h T t − t 0 = u 0 T ( log cosh T t − t 0 − log cosh T t 0 )
= u 0 T log cosh t − t 0 T cosh t 0 T =u_0T\log\frac{\cosh\frac{t-t_0}{T}}{\cosh\frac{t_0}{T}} = u 0 T log c o s h T t 0 c o s h T t − t 0 , if u 0 > v 0 u_0>v_0 u 0 > v 0
If u 0 < v 0 u_0<v_0 u 0 < v 0 then s ( t ) = s ( 0 ) + ∫ 0 t v ( t ) d t = ∫ 0 t u 0 coth t − t 0 T d t = s(t)=s(0)+\int\limits_{0}^{t}v(t)dt=\int\limits_{0}^{t}u_0\coth\frac{t-t_0}{T}dt= s ( t ) = s ( 0 ) + 0 ∫ t v ( t ) d t = 0 ∫ t u 0 coth T t − t 0 d t =
= u 0 T ∫ 0 t d sinh t − t 0 T sinh t − t 0 T = u 0 T ( log sinh t − t 0 T − log sinh t 0 T ) =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}) = u 0 T 0 ∫ t s i n h T t − t 0 d s i n h T t − t 0 = u 0 T ( log sinh T t − t 0 − log sinh T t 0 )
= u 0 T log sinh t − t 0 T sinh t 0 T =u_0T\log\frac{\sinh\frac{t-t_0}{T}}{\sinh\frac{t_0}{T}} = u 0 T log s i n h T t 0 s i n h T t − t 0
If u 0 = v 0 u_0=v_0 u 0 = v 0 then s ( t ) = s ( 0 ) + ∫ 0 t v ( t ) d t = ∫ 0 t u 0 d t = u 0 t s(t)=s(0)+\int\limits_{0}^{t}v(t)dt=\int\limits_{0}^{t}u_0dt=u_0t s ( t ) = s ( 0 ) + 0 ∫ t v ( t ) d t = 0 ∫ t u 0 d t = u 0 t
Answer .
if u 0 > v 0 u_0>v_0 u 0 > v 0 then
(a) v ( t ) = u 0 tanh t − t 0 T v(t)=u_0\tanh\frac{t-t_0}{T} v ( t ) = u 0 tanh T t − t 0 , where u 0 = m g k u_0=\sqrt\frac{mg}{k} u 0 = k m g , T = u 0 g = m k g T=\frac{u_0}{g}=\sqrt\frac{m}{kg} T = g u 0 = k g m and t 0 = − 1 2 log u 0 + v 0 u 0 − v 0 t_0=-\frac{1}{2}\log\frac{u_0+v_0}{u_0-v_0} t 0 = − 2 1 log u 0 − v 0 u 0 + v 0
(b) lim t → + ∞ v ( t ) = m g k \lim\limits_{t\to+\infty}v(t)=\sqrt\frac{mg}{k} t → + ∞ lim v ( t ) = k m g
(c) s ( t ) = u 0 T log cosh t − t 0 T cosh t 0 T s(t)=u_0T\log\frac{\cosh\frac{t-t_0}{T}}{\cosh\frac{t_0}{T}} s ( t ) = u 0 T log c o s h T t 0 c o s h T t − t 0
if u 0 < v 0 u_0<v_0 u 0 < v 0 then
(a) v ( t ) = u 0 coth t − t 0 T v(t)=u_0\coth\frac{t-t_0}{T} v ( t ) = u 0 coth T t − t 0 , where u 0 = m g k u_0=\sqrt\frac{mg}{k} u 0 = k m g , T = u 0 g = m k g T=\frac{u_0}{g}=\sqrt\frac{m}{kg} T = g u 0 = k g m and t 0 = − 1 2 log v 0 + u 0 v 0 − u 0 t_0=-\frac{1}{2}\log\frac{v_0+u_0}{v_0-u_0} t 0 = − 2 1 log v 0 − u 0 v 0 + u 0
(b) lim t → + ∞ v ( t ) = m g k \lim\limits_{t\to+\infty}v(t)=\sqrt\frac{mg}{k} t → + ∞ lim v ( t ) = k m g
(c) s ( t ) = u 0 T log sinh t − t 0 T sinh t 0 T s(t)=u_0T\log\frac{\sinh\frac{t-t_0}{T}}{\sinh\frac{t_0}{T}} s ( t ) = u 0 T log s i n h T t 0 s i n h T t − t 0
if u 0 = v 0 u_0=v_0 u 0 = v 0 then
(a) v ( t ) = v 0 = u 0 v(t)=v_0=u_0 v ( t ) = v 0 = u 0
(b) lim t → + ∞ v ( t ) = m g k \lim\limits_{t\to+\infty}v(t)=\sqrt\frac{mg}{k} t → + ∞ lim v ( t ) = k m g ,
(c) s ( t ) = u 0 t s(t)=u_0t s ( t ) = u 0 t
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