Impact Mechanics – Loss of Kinetic Energy in a Perfectly Inelastic Collision Two bodies of masses m1 and m2 move along the same line with velocities u1 and u2, respectively, and stick together after impact (perfectly inelastic). What is the loss of kinetic energy?

Introduction / Context:
In one-dimensional inelastic impact where bodies coalesce, momentum is conserved but kinetic energy is not. The classic result expresses the energy loss in terms of masses and the initial relative speed.

Given Data / Assumptions:

• Masses: m1 and m2.
• Initial velocities: u1 and u2 along the same line.
• Perfectly inelastic collision: common final velocity v after impact.
• No external impulse along line of motion.

Concept / Approach:
Use conservation of linear momentum to find v, then compute initial and final kinetic energies and subtract to get the loss ΔKE.

Step-by-Step Solution:
Momentum conservation: (m1 * u1 + m2 * u2) = (m1 + m2) * v ⇒ v = (m1 u1 + m2 u2) / (m1 + m2). Initial KE: KE_i = (1/2) m1 u1^2 + (1/2) m2 u2^2. Final KE: KE_f = (1/2) (m1 + m2) v^2. Subtract: ΔKE = KE_i − KE_f = (m1 * m2 * (u1 − u2)^2) / (2 * (m1 + m2)).

Verification / Alternative check:
The loss is proportional to the square of relative speed (u1 − u2); if u1 = u2, there is no impact and ΔKE = 0.

Why Other Options Are Wrong:
(b) misses the mass product and overestimates energy loss. (c) uses sum of speeds, not relative speed. (d) true only if u1 = u2; generally false. (e) dimensionally incorrect.

Common Pitfalls:
Forgetting to square the relative speed or misapplying momentum conservation.

Final Answer:
ΔKE = (m1 * m2 * (u1 − u2)^2) / (2 * (m1 + m2))