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Question Number 50914 by peter frank last updated on 22/Dec/18
prove that relative velocity is reversed by a head on collision
$${prove}\:{that}\:{relative}\:{velocity} \\ $$$${is}\:{reversed}\:{by}\:{a}\:{head}\:{on} \\ $$$${collision} \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 22/Dec/18
for collission velocity are v_1 ^→ =u_1 i^→ andv_2 ^⌣ = −u_2 i^→ relative velocity of v_1 ^→ w.r.t v_2 ^→ is=v_1 ^→ −v_2 ^→ =u_1 i^→ +u_2 i^→ so velocity v_2 ^→ directiin reversed and moving along +ve x axis...
$${for}\:{collission}\:\:{velocity}\:{are}\:\overset{\rightarrow} {{v}}_{\mathrm{1}} ={u}_{\mathrm{1}} \overset{\rightarrow} {{i}}\:\:{and}\overset{\smile} {{v}}_{\mathrm{2}} =\:−{u}_{\mathrm{2}} \overset{\rightarrow} {{i}} \\ $$$${relative}\:{velocity}\:{of}\:\overset{\rightarrow} {{v}}_{\mathrm{1}} \:{w}.{r}.{t}\:\overset{\rightarrow} {{v}}_{\mathrm{2}} \:{is}=\overset{\rightarrow} {{v}}_{\mathrm{1}} −\overset{\rightarrow} {{v}}_{\mathrm{2}} \\ $$$$={u}_{\mathrm{1}} \overset{\rightarrow} {{i}}+{u}_{\mathrm{2}} \overset{\rightarrow} {{i}} \\ $$$${so}\:{velocity}\:\overset{\rightarrow} {{v}}_{\mathrm{2}} \:{directiin}\:{reversed}\:{and}\:{moving} \\ $$$${along}\:+{ve}\:{x}\:{axis}… \\ $$
Commented by peter frank last updated on 22/Dec/18
thank you
$${thank}\:{you} \\ $$
Answered by peter frank last updated on 22/Dec/18
from conservation of linear momentum m_1 u_(1 ) +m_(2 ) u_2 =m_1 v_1 +m_(2 ) v_2 m_(1 ) (u_(1 ) −v_(1 ) )=m_2 (v_(2 ) −u_2 )......(i) from K.E conservation (1/2)m_1 u_1 ^2 +(1/2)m_2 u_2 ^2 =(1/2)m_(1 ) v_(1 ) ^2 +(1/2)m_2 v_(2 ) ^2 m_1 (u_(1 ) ^2 −v_(1 ) ^2 )=m_(2 ) (v_(2 ) ^2 −u_2 ^2 ).....(ii) divide eqn (ii) by eqn (i) u_1 +v_(1 ) =v_2 +u_2 u_1 −u_(2 ) =v_(2 ) −v_1 u_2 −u_1 =−(v_2 −v_1 )
$${from}\:{conservation}\:{of}\: \\ $$$${linear}\:{momentum} \\ $$$${m}_{\mathrm{1}} {u}_{\mathrm{1}\:} +{m}_{\mathrm{2}\:} {u}_{\mathrm{2}} ={m}_{\mathrm{1}} {v}_{\mathrm{1}} \:+{m}_{\mathrm{2}\:\:} {v}_{\mathrm{2}} \\ $$$${m}_{\mathrm{1}\:} \left({u}_{\mathrm{1}\:} −{v}_{\mathrm{1}\:} \right)={m}_{\mathrm{2}} \left({v}_{\mathrm{2}\:} −{u}_{\mathrm{2}} \right)……\left({i}\right) \\ $$$${from}\:{K}.{E}\:{conservation} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}{m}_{\mathrm{1}} {u}_{\mathrm{1}} ^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{2}}{m}_{\mathrm{2}} {u}_{\mathrm{2}} ^{\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{2}}{m}_{\mathrm{1}\:} {v}_{\mathrm{1}\:} ^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{2}}{m}_{\mathrm{2}} {v}_{\mathrm{2}\:} ^{\mathrm{2}} \\ $$$${m}_{\mathrm{1}} \left({u}_{\mathrm{1}\:} ^{\mathrm{2}} −{v}_{\mathrm{1}\:\:} ^{\mathrm{2}} \right)={m}_{\mathrm{2}\:} \left({v}_{\mathrm{2}\:\:} ^{\mathrm{2}} −{u}_{\mathrm{2}} ^{\mathrm{2}} \right)…..\left({ii}\right) \\ $$$${divide}\:{eqn}\:\:\left({ii}\right)\:{by}\:\:{eqn}\:\left({i}\right) \\ $$$${u}_{\mathrm{1}} +{v}_{\mathrm{1}\:} ={v}_{\mathrm{2}} +{u}_{\mathrm{2}} \\ $$$${u}_{\mathrm{1}} −{u}_{\mathrm{2}\:} ={v}_{\mathrm{2}\:\:\:} −{v}_{\mathrm{1}} \\ $$$${u}_{\mathrm{2}} −{u}_{\mathrm{1}} =−\left({v}_{\mathrm{2}} −{v}_{\mathrm{1}} \right) \\ $$

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