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Question-138062




Question Number 138062 by I want to learn more last updated on 09/Apr/21
Answered by mr W last updated on 09/Apr/21
(i)  (a)  S: M_1 =4m  T: M_2 =m  v_(1i) =u  v_(2f) −v_(1f) =ev_(1i)   M_2 v_(2f) +M_1 v_(1f) =M_1 v_(1i)   v_(2f) +4v_(1f) =4v_(1i)   v_(1f) +ev_(1i) +4v_(1f) =4v_(1i)   5v_(1f) =(4−e)v_(1i)   v_(1f) =(((4−e)u)/5)  v_(2f) =(((4−e)u)/5)+eu=((4(1+e)u)/5)  (b)  v_(1f,max) =((4u)/5) when e=0  v_(1f,min) =((3u)/5) when e=1  (c)  M_1 (v_(1i) −v_(1f) )=(3/(10))M_1 v_(1i)   v_(1f) =((7v_(1i) )/(10))  (((4−e)u)/5)=((7u)/(10))  4−e=(7/2)  ⇒e=(1/2)
$$\left({i}\right) \\ $$$$\left({a}\right) \\ $$$${S}:\:{M}_{\mathrm{1}} =\mathrm{4}{m} \\ $$$${T}:\:{M}_{\mathrm{2}} ={m} \\ $$$${v}_{\mathrm{1}{i}} ={u} \\ $$$${v}_{\mathrm{2}{f}} −{v}_{\mathrm{1}{f}} ={ev}_{\mathrm{1}{i}} \\ $$$${M}_{\mathrm{2}} {v}_{\mathrm{2}{f}} +{M}_{\mathrm{1}} {v}_{\mathrm{1}{f}} ={M}_{\mathrm{1}} {v}_{\mathrm{1}{i}} \\ $$$${v}_{\mathrm{2}{f}} +\mathrm{4}{v}_{\mathrm{1}{f}} =\mathrm{4}{v}_{\mathrm{1}{i}} \\ $$$${v}_{\mathrm{1}{f}} +{ev}_{\mathrm{1}{i}} +\mathrm{4}{v}_{\mathrm{1}{f}} =\mathrm{4}{v}_{\mathrm{1}{i}} \\ $$$$\mathrm{5}{v}_{\mathrm{1}{f}} =\left(\mathrm{4}−{e}\right){v}_{\mathrm{1}{i}} \\ $$$${v}_{\mathrm{1}{f}} =\frac{\left(\mathrm{4}−{e}\right){u}}{\mathrm{5}} \\ $$$${v}_{\mathrm{2}{f}} =\frac{\left(\mathrm{4}−{e}\right){u}}{\mathrm{5}}+{eu}=\frac{\mathrm{4}\left(\mathrm{1}+{e}\right){u}}{\mathrm{5}} \\ $$$$\left({b}\right) \\ $$$${v}_{\mathrm{1}{f},{max}} =\frac{\mathrm{4}{u}}{\mathrm{5}}\:{when}\:{e}=\mathrm{0} \\ $$$${v}_{\mathrm{1}{f},{min}} =\frac{\mathrm{3}{u}}{\mathrm{5}}\:{when}\:{e}=\mathrm{1} \\ $$$$\left({c}\right) \\ $$$${M}_{\mathrm{1}} \left({v}_{\mathrm{1}{i}} −{v}_{\mathrm{1}{f}} \right)=\frac{\mathrm{3}}{\mathrm{10}}{M}_{\mathrm{1}} {v}_{\mathrm{1}{i}} \\ $$$${v}_{\mathrm{1}{f}} =\frac{\mathrm{7}{v}_{\mathrm{1}{i}} }{\mathrm{10}} \\ $$$$\frac{\left(\mathrm{4}−{e}\right){u}}{\mathrm{5}}=\frac{\mathrm{7}{u}}{\mathrm{10}} \\ $$$$\mathrm{4}−{e}=\frac{\mathrm{7}}{\mathrm{2}} \\ $$$$\Rightarrow{e}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$
Commented by I want to learn more last updated on 10/Apr/21
God bless you sir. I appreciate.
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}.\:\mathrm{I}\:\mathrm{appreciate}. \\ $$
Answered by mr W last updated on 10/Apr/21
(ii)  v_i =30i  m(v_f −v_i )=2i−10j  ⇒v_f =v_i +10i−50j=40i−50j
$$\left({ii}\right) \\ $$$$\boldsymbol{{v}}_{\boldsymbol{{i}}} =\mathrm{30}{i} \\ $$$${m}\left(\boldsymbol{{v}}_{\boldsymbol{{f}}} −\boldsymbol{{v}}_{\boldsymbol{{i}}} \right)=\mathrm{2}{i}−\mathrm{10}{j} \\ $$$$\Rightarrow\boldsymbol{{v}}_{\boldsymbol{{f}}} =\boldsymbol{{v}}_{\boldsymbol{{i}}} +\mathrm{10}{i}−\mathrm{50}{j}=\mathrm{40}{i}−\mathrm{50}{j} \\ $$
Commented by I want to learn more last updated on 10/Apr/21
Thanks sir. I appreciate.
$$\mathrm{Thanks}\:\mathrm{sir}.\:\mathrm{I}\:\mathrm{appreciate}. \\ $$

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