Question

Explain perfect inelastic collision and derive an expression for loss of kinetic energy in perfect inelastic collision?

Answer 1

In a perfectly inelastic or completely inelastic collision, the objects stick together permanently after collision such that they move with common velocity. Let the two bodies with masses m₁ and m₂ move with initial velocities u₁ and u₂ respectively before collision. After perfect inelastic collision both the objects move together with a common velocity v as shown in figure.

Since, the linear momentum is conserved during collisions,

m₁ u₁ + m₂ u₂ = (m₁ + m₂) v

The common velocity can be computed by

V = \(\frac{m_1u_1 + m_2 u_2}{(m_1 + m_2)}\)  .....(1)

Loss of kinetic energy in perfect inelastic collision; In perfectly inelastic collision, the loss in kinetic energy during collision is transformed to another form of energy like sound, thermal, heat, light etc. Let KE_i be the total kinetic energy before collision and KE_f be the total kinetic energy after collision. Total kinetic energy before collision,

Total kinetic energy after collision,

Then the loss of kinetic energy is Loss of KE, ∆Q = KE_f - KE_i

Substituting equation (1) in equation (4), and on simplifying (expand v by using the algebra (a + b)² = a² + b² + 2ab), we get

Loss of KE, ∆Q = \(\frac{1}{2}\) \((\frac{m_1 + m_2}{m_1 + m_2})\) (u₁ - u₂)²