## Abhisar one year ago 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.

1. 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.

2. 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, $$\sf \boxed{{V_b}^{'}=\frac{2M_1V_1}{M_1+M_2}}$$

3. IrishBoy123
4. IrishBoy123

and the "ie" here is a non sequitur https://gyazo.com/64cbadb3997368b8e712800f0163f5c7 because it confuses/conflates conservation of momentum with conservation of energy

5. Abhisar

Thanks @irishboy123 , It should be, $$\sf Va+{V_a}^{'}={V_b}^{'}$$

6. arindameducationusc

yes, even I was wondering .... Thanks to @irishboy123 And Awesome derivation Abhisar, It was very useful and got a good revision. Thank you

7. Abhisar

I am glad you found it helpful c: