Consider two elastic bodies of masses `m_(1)` and `m_(2)` 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_(1) u_(1) + m_(2) u_(2) = m_(1)v_(1) + m_(2)v_(2)` ...(1)
or `m_(1)(u_(1) - v_(1)) = m_(2)(v_(2) - u_(2))` ...(2)
Father,
For elastic collision,
Total kinetic energy before collision `KE_(i)` = Total kinetic energy after collision `KE_(f)`
`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)` ...(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_(2) + 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_(2)`
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 inal velocities `v_(1)` and `v_(2)`:
Substituting equation (7) 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_(1)u_(1) - m_(1) v_(1) = m_(2)u_(1) + m_(2)v_(1) + 2m_(2)u_(2)`
`m_(1)u_(1) - m_(2) u_(1) + 2 m_(2) u_(2) = m_(1) v_(1) + m_(2) v_(1)`
`(m_(1) - m_(2)) u_(1) + 2m_(2)u_(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)` ...(8)
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)
