Key Applications

1. Free Fall (from height H): $v = \sqrt{2gH}$ (All PE converts to KE at ground level)

2. Projectile (thrown horizontally at speed u from height H): $v_{\text{ground}} = \sqrt{u^2 + 2gH}$ (Initial KE = ½$mu^{2}$; gained PE = mgH converts to KE)

3. Pendulum (amplitude h):

• At extreme: KE = 0, PE = mgh (maximum)
• At mean: KE = mgh (maximum), PE = 0 $v_{\text{mean}} = \sqrt{2gh}$

4. Spring-Block (compressed by x_{0}, frictionless): $v_{\text{natural length}} = x_0\sqrt{\frac{k}{m}}$

5. Roller Coaster (starts at rest, height H; speed at height h): $v = \sqrt{2g(H-h)}$

When Energy Conservation Fails