A particle of mass m₁ moving with u₁ velocity collide with another particle of mass m₂ which is initially in rest. If collision is perfectly inelastic then the fractional loss in the kinetic energy is

CONCEPT:

• A perfectly elastic collision is defined as one in which there is no loss of kinetic energy in the collision.
• An inelastic collision is one in which part of the kinetic energy is changed to some other form of energy in the collision.

EXPLANATION:

• By the law of conservation of momentum

⇒ m₁u₁ = (m₁ + m₂)V

\(\Rightarrow V = \frac{m_1u_1}{m_1+m_2}\)  

• By the law of conservation of kinetic energy

\(\Rightarrow \frac{1}{2}m_1u^2_1=\frac{1}{2}(m_1+m_2)V^2\)

• The ratio of kinetic energy before and after the collision is 

\(\frac{KE_f}{KE_i}=\frac{\frac{1}{2}(m_1+m_2)V^2}{\frac{1}{2}mu^2_1}=\frac{\frac{1}{2}(m_1+m_2)[\frac{m_1u_1}{m_1+m_2}]^2}{\frac{1}{2}mu^2_1}=\frac{m_1}{m_1+m_2}\)

• The fractional loss in the kinetic energy is

\(\Rightarrow \frac{KE_i-KE_f}{KE_i}=\frac{KE_i([1-\frac{KE_f}{KE_i}])}{KE_i}=\frac{KE_i([1-\frac{m_1}{m_1+m_2}])}{KE_i}=\frac{m_2}{m_1+m_2}\)