Two particles each of mass m and travelling with velocities...

Two particles each of mass $m$ and travelling with velocities $u_1$ and $u_2$ collide perfectly inelastically. The loss of energy will be (a) $\frac{1}{4} m\left(u_1-u_2\right)^2$ (b) $\frac{1}{2} m\left(u_1-u_2\right)^2$ (c) $2 m\left(u_1-u_2\right)^2$ (d) $m\left(\mathrm{u}_1-u_2\right)^2$

Two particles each of mass $m$ and travelling with velocities $u_1$ and $u_2$ collide perfectly inelastically. The loss of energy will be
(a) $\frac{1}{4} m\left(u_1-u_2\right)^2$
(b) $\frac{1}{2} m\left(u_1-u_2\right)^2$
(c) $2 m\left(u_1-u_2\right)^2$
(d) $m\left(\mathrm{u}_1-u_2\right)^2$

Key Concepts

Relative Velocity

The concept of relative velocity involves the difference in velocities between two objects. In collision problems, particularly when determining energy loss, the square of the difference in velocities often appears in formulas, since kinetic energy depends on the square of the velocity. Relative velocity provides insight into the energy dissipated during the collision.

Kinetic Energy Loss

Kinetic energy loss in inelastic collisions is the difference between the initial kinetic energy of the system and the final kinetic energy after the collision. This loss, which occurs due to the transformation of some kinetic energy into other energy forms, is a key factor in analyzing the outcomes of inelastic collisions.

Conservation of Momentum

The law of conservation of momentum states that the total momentum of a closed system remains constant provided no external forces act on it. This principle is crucial for determining the final velocity of the combined mass in inelastic collisions since despite energy losses, the momentum before and after the collision must be equal.

Perfectly Inelastic Collision

In a perfectly inelastic collision, the colliding objects stick together after impact, meaning they move as a single combined mass. Although momentum is conserved in these collisions, kinetic energy is not; some of the initial kinetic energy is transformed into other forms of energy, such as heat or deformation energy.

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