Collisions - Loss of Kinetic Energy in Perfectly Inelastic Head-On Collision

When two objects collide and stick together, it is called a perfectly inelastic collision. In this type of collision, the two bodies move together as one unit after the impact. Although momentum is conserved during the collision, kinetic energy is not fully conserved. Some kinetic energy is always lost and converted into other forms of energy, like heat, sound, or deformation. Understanding this loss helps us analyze real-world collisions accurately.

Perfectly Inelastic Collision: A perfectly inelastic, head-on collision of two bodies of masses m₁ and m₂ with respective initial velocities u₁ and u₂, where they move jointly after the collision, i.e., their final velocity is the same.

Initial momentum = Final momentum

m₁u₁ + m₂u₂ = (m₁ + m₂)v

Solve for the common final velocity v:

\[v=\frac{m_1u_1+m_2u_2}{m_1+m_2}\]

This gives us the velocity with which both bodies move together after the perfectly inelastic collision.

Loss in K.E. = Total initial K.E. - Total final K.E.

\[\frac{1}{2}m_1u_1^2+\frac{1}{2}m_2u_2^2\]

\[\frac{1}{2}(m_1+m_2)v^2\]

\[\Delta(K.E.)=\frac{1}{2}m_1u_1^2+\frac{1}{2}m_2u_2^2-\frac{1}{2}(m_1+m_2)\left(\frac{m_1u_1+m_2u_2}{m_1+m_2}\right)^2\]

\[\Delta(K.E.)=\frac{1}{2}\left(\frac{m_1m_2}{m_1+m_2}\right)(u_1-u_2)^2\]

Since masses are always positive and (u₁ − u₂)² is also positive, there is always a loss in kinetic energy in a perfectly inelastic collision.

For an inelastic collision with a coefficient of restitution e, the final velocities are:

\[v_1=\left(\frac{m_1-em_2}{m_1+m_2}\right)u_1+\left(\frac{[1+e]m_2}{m_1+m_2}\right)u_2\]
or
\[v_1=\frac{em_2(u_2-u_1)+m_1u_1+m_2u_2}{m_1+m_2}\]

Similarly, for the second body:
\[v_2=\left(\frac{m_2-em_1}{m_1+m_2}\right)u_2+\left(\frac{[1+e]m_1}{m_1+m_2}\right)u_1\]
or
\[v_2=\frac{em_1(u_1-u_2)+m_1u_1+m_2u_2}{m_1+m_2}\]

\[\Delta(K.E.)=\frac{1}{2}\left(\frac{m_1m_2}{m_1+m_2}\right)(u_1-u_2)^2(1-e^2)\]

Since $e<\mathrm{1,}$the term (1 − e²) is always positive, there is always a loss of kinetic energy in an inelastic collision.

For a perfectly inelastic collision, $e=0$, and the loss is maximum.

The quantity μ = \[\frac{m_1m_2}{m_1+m_2}\] is called the reduced mass of the system.

During a collision, the linear momentum delivered by the first body to the second body is equal to the change in momentum of the second body, and vice versa.

After substituting the values of v₁ and v₂ and solving:
\[|J|=\left(\frac{m_1m_2}{m_1+m_2}\right)(1+e)(|u_1-u_2|)\]
or
\[|J|=\mu(1+e)u_{relative}\]
where |u₁ - u₂| = u_relative is the velocity of approach.

• Energy transformation: Shows how kinetic energy is converted to other forms during a collision
• Maximum loss occurs: In perfectly inelastic collisions ($e=0$), the loss of kinetic energy is maximum
• Momentum conservation: Even though kinetic energy is lost, linear momentum is always conserved
• Reduced mass concept: Introduces the reduced mass μμ, which simplifies collision calculations
• Universal application: These formulas work for any two-body collision system
• Coefficient role: The coefficient of restitution e determines how much energy is lost
• Impulse equality: Both colliding bodies experience equal impulse but in opposite directions

• Car accidents: When two cars collide and their bumpers get stuck together, they move as one unit after impact. This is a perfectly inelastic collision where maximum kinetic energy is lost as sound, heat, and deformation of metal.
• Clay balls collision: When two clay balls are thrown at each other, they stick together upon impact and move with a common velocity, losing kinetic energy to deformation.
• Bullet embedding in wood: When a bullet is fired into a wooden block and gets embedded in it, both move together afterward. The lost kinetic energy converts to heat and sound.
• Railway coupling: When a moving railway wagon couples with a stationary wagon, they lock together and move with reduced velocity, demonstrating a perfectly inelastic collision.
• Tackling in sports: In rugby or football, when a player tackles another and both fall together, it resembles a perfectly inelastic collision where momentum is conserved but kinetic energy is lost.
• Meteorite impact: When a meteorite strikes the Earth and embeds itself, it's a perfectly inelastic collision where enormous kinetic energy converts to heat, sound, and crater formation.