Abhisar
  • Abhisar
I was teaching a kid about Elastic Head on collision and he was having some trouble deriving few relations so I am doing this post to help him and others looking for similar content. \(\bigstar\) There is a typographical mistake below in Eq.3. The correct equation should be \(\sf Va+{V_a}^{'}={V_b}^{'}\) \(\bigstar\) There is one another typographical mistake in the statement just below Eq.1. The correct statement should be: Also, in both elastic or inelastic collision momentum is conserved i.e. total initial momentum of the system is equal to the total final momentum of the system.
Physics
jamiebookeater
  • jamiebookeater
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Abhisar
  • Abhisar
Let's suppose that a body of mass \(\sf M_a\) is travelling with a velocity \(\sf V_a\) along a straight line and it collides with another stationary body of mass \(\sf M_b\). Consider the collision to be elastic in nature.
Abhisar
  • Abhisar
\(\huge \bigstar\) Derive an equation for final velocities \(\sf V_a^{'}~and~V_b^{'}\) in terms of \(\sf M_a, M_b~and~V_a\). Since, the collision is elastic we can say that final kinetic energy of the system is equal to the initial kinetic energy. \(\sf \Rightarrow \frac{1}{2}{M_aV_a}^2 = \frac{1}{2}M_a{V_a^{'}}^2+\frac{1}{2}M_b{V_b^{'}}^2\) \(\sf \Rightarrow {M_a(V_a}^{2}-{V_a^{'}}^2)=M_b{V_b^{'}}^2\) \(\Rightarrow \sf M_a(V_a-V_a^{'})(V_a+V_a^{'})=M_b{V_b^{'}}^2\) ....Eq.1 Also, in both elastic or inelastic collision momentum is conserved i.e. total initial energy of the system is equal to the total final energy of the system. \(\sf \Rightarrow M_aV_a=M_aV_a^{'}+M_bV_b^{'}\) \(\sf \Rightarrow M_a(V_a-V_a^{'})=M_bV_b^{'}\) ...........Eq.2 Dividing Eq.1 with Eq.2 we get, \(\sf V_a+{V_b}^{'}={V_b}^{'}\) .........Eq.3 Substituting this value in Eq.2 we get, \(\boxed{\sf {V_a}^{'}=\frac{V_a(M_a-M_b)}{M_a+M_b}}\) Substituting this value in Eq.3 we get, \(\sf \boxed{{V_b}^{'}=\frac{2M_1V_1}{M_1+M_2}}\)
IrishBoy123
  • IrishBoy123
https://gyazo.com/e1ca809cd98628a8b98808eff6b99869 just a typo

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IrishBoy123
  • IrishBoy123
and the "ie" here is a non sequitur https://gyazo.com/64cbadb3997368b8e712800f0163f5c7 because it confuses/conflates conservation of momentum with conservation of energy
Abhisar
  • Abhisar
Thanks @irishboy123 , It should be, \(\sf Va+{V_a}^{'}={V_b}^{'}\)
arindameducationusc
  • arindameducationusc
yes, even I was wondering .... Thanks to @irishboy123 And Awesome derivation Abhisar, It was very useful and got a good revision. Thank you
Abhisar
  • Abhisar
I am glad you found it helpful c:

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