Work, Energy & Power — Collisions, Springs & Power (Medium)

Quick Recap — Collisions, Springs & Power

• Collisions: momentum is conserved in all; kinetic energy is conserved only in elastic ones. Perfectly inelastic: bodies stick, $v=\dfrac{m_1u_1+m_2u_2}{m_1+m_2}$.
• 1D elastic (equal masses): the two velocities are simply exchanged. Coefficient of restitution $e=\dfrac{\text{speed of separation}}{\text{speed of approach}}$.
• Bouncing ball: after each bounce the rebound height is $e^2$ times the previous ($h_n=e^{2n}h_0$).
• Spring launch: $\tfrac12 kx^2=\tfrac12 mv^2$. Pump power raising and ejecting water: $P=\dfrac{dm}{dt}\left(gh+\tfrac12 v^2\right)$.

Worked mini-example. A ball dropped from 2 m rebounds to 0.5 m: $e=\sqrt{\dfrac{h'}{h}}=\sqrt{\dfrac{0.5}{2}}=0.5$.