Elastic collision: In a collision, the total initial kinetic energy of the bodies (before collision) is not equal to the total final kinetic energy of the bodies (after collision) then, it is called as inelastic collision. i.e.,
Consider two elastic bodies of masses m, and m, moving in a straight line (along positive x direction) on a frictionless horizontal surface as shown in figure.
In order to have collision, we assume that the mass `m_1 `moves faster than mass `m_2 i.e., u_1 gt u_2`. For elastic collision, the total linear momentum and kinetic energies of the two bodies before and after collision must remain the same.
From the law of conservation of linear momentum, Total momentum before collision `(P_i)` = Total momentum after collision `(P_f)`
`m_1u_1 + m_2u_2 = m_1v_1 + m_2v_1 ` ...(1)
`m_1(u_1 - v_1) - m_2(v_2 - u_2)` ......(2)
further
For elastic collision, Total kinetic energy before collision `KE_i` = Total kinetic energy after collision `KE_f`
`1/2 m_1u_1^2 + 1/2 m_2u_2^2 = 1/2 m_1v_1^2 + 1/2 m_2v_2^2` ....(3)
After simplifying and rearranging the terms,
`m_1 (u_1^2 -v_1^2) = m_2 (v_2^2 - u_2^2)`
Using the formula ` a^2 -b^2 = (a + b) (a - b) ` , we can rewrite the above equation as
`m_1 (u_1 + v_1) (u_1- v_1) = m_2 (v_1 + u_2) (v_2 - u_2)` ....(4)
Dividing equation (4) by (2) gives,
` (m_1(u_1 + v_1)(u_1 - v_1) )/(m_1 (u_1 - v_1)) = (m_2(v_2 + u_2)(v_2-u_2) )/(m_2(v_2 - u_2) )`
`u_1 + v_1 = v_2 +u_1`
Rearranging `u_1 - u_2 =v_2 - v_1` .....(5)
Equation (5) can be rewritten as
`u_1 - u_2 = - (v_1 - v_2)`
This means that for any elastic head on collision, the relative speed of the two elastic bodies after the collision has the same magnitude as before collision but in opposite direction. Further note that this result is independent of mass. Rewriting the above equation for `v_1` and `v_2`
`v_1 = v_2 + u_2 -u_1` ....(6)
or `v_2 = u_1 + v_1 - u_1` .....(7)
To find the final velocities `v_1` and `v_2` :
Substituting equation (5) in equation (2) gives the velocity of `m_1` as
`m_1 (u_1 - v_1) = m_2 (u_1 + v_1 - u_2 - u_2)`
`m_1 (u_1 - v_1) = m_2 (u_1 + v_1 - 2u_2)`
`m_1u_1 - m_1v_1 = m_2u_1 + m_2v_1 + 2m_2u_2`
`m_1u_1 - m_2u_1 + 2m_2u_2 = m_1 v_1 + m_2v_1`
(`m_1 - m_2) u_1 + 2m_2u_2 = (m_1 + m_2) v_1`
or ` v_1 = ( (m_1 - m_2)/(m_1 + m_2) ) u_1 + ( (2m_2)/(m_1 + m_2) ) u_2`
Similarly, by substituting (6) in equation (2) or substituting equation (8) in equation (7), we get the final velocity of `m_2` as
` v_2 = ( (2m_1)/(m_1 + m_2)) u_1 + ((m_2 - m_1)/(m_1 + m_2) ) u_2` .....(9)
