Work, Energy & Power — Advanced Multi-Step Practice (Hard)

Quick Recap — Advanced Energy & Collisions (Multi-step)

• Chained conservation: momentum through a collision, then energy after (ballistic pendulum: $v=\dfrac{m}{m+M}u$, then $h=\dfrac{v^2}{2g}$).
• 1D elastic (general): $v_1'=\dfrac{(m_1-m_2)u_1}{m_1+m_2}$, $v_2'=\dfrac{2m_1 u_1}{m_1+m_2}$ (target at rest).
• Bouncing ball total path: $H_{total}=h\dfrac{1+e^2}{1-e^2}$; total time $\dfrac{2v_0}{g}\dfrac{1}{1-e}$-type series.
• Loop-the-loop: minimum release height on a smooth track to complete a loop of radius $r$ is $2.5r$.
• Spring + friction: equate the spring's stored energy to $\tfrac12 mv^2$ plus friction work over the slide.

Worked mini-example. A 10 g bullet at 400 m/s embeds in a 1.99 kg block: $v=\dfrac{0.01\times400}{2.0}=2$ m/s, and the block rises $h=\dfrac{v^2}{2g}=\dfrac{4}{20}=0.2$ m.