Question

Two objects move towards each other along a straight line and collide inelastically. It is known that the ratio of their kinetic energies is n. What should be the ratio of their massess fo the objects to move after the collision in the direction of motion of the object with the smallest kinetic energy?

Please explain solutions.

Answer 1

Let K.E 1 > K.E 2

or, 0.5 m1 * v1² > 0.5 m2 * v2² ?

or, m1/ m2 > v2² ?/ v1²

K.E 1 / K.E 2 = 0.5 m1 * v1² /0.5 m2 * v2² ?

or, n = 0.5 m1 * v1² /0.5 m2 * v2² ?

or, n = m1 * v1² / m2 * v2² ?

or, v2² / v1² = m1/m2 * 1/n ...............(1)

From Conservation of momentum

m1 (- v1) + m2 v2 = (m1 + m2) v

or, - m1 v1 / m2 + v2 = (m1/m2 + 1) v

or, { (- m1 v1 / m2) + v2 } / (m1/m2 + 1) = v > 0 (since, denominator is + ve, so numerator must be +ve)

or, { (- m1 v1 / m2) + v2 } > 0

or, - m1 / m2 > - v2 / v1

or, m1 /m2 < v2/v1

or, m1 / m2 < sqrt ( m1/m2 * n) using equation ....(1)

or, sqrt (m1 / m2 ) < sqrt ( 1/n)

or, m1/m2 < 1/n