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A particle is projected vertically upward with $u = 10\text{ m/s}$. Air exerts a resistive force $F = -0.2v^2$ on it ($m = 2\text{ kg}$, $g = 10\text{ m/s}^2$). The maximum height attained is:
Net force (taking up as positive): $F_{net} = -mg - 0.2v^2 = -20 - 0.2v^2$
Using $ma = v\dfrac{dv}{dx}$: $2v\dfrac{dv}{dx} = -20 - 0.2v^2$
Separating variables: $\dfrac{v\,dv}{20+0.2v^2} = -\dfrac{dx}{2}$
Integrating from $v = 10$ to $v = 0$: let $k = 20 + 0.2v^2$, $dk = 0.4v\,dv$
$\dfrac{1}{0.4}\big[\ln(20) - \ln(40)\big] = -\dfrac{H}{2} \Rightarrow H = \dfrac{2\ln 2}{0.4} = \mathbf{5\ln 2}$
A cantilever is a beam or structural element that:
- Is fixed (anchored) at one end only
- The other end projects freely without support
- Carries load primarily through bending (hogging moment at the fixed end)
Famous cantilever examples in architecture:
- Fallingwater (Frank Lloyd Wright) — dramatic cantilevered terraces over a waterfall
- CN Tower observation deck
- Balconies, overhanging roof eaves, diving boards
The tension face of a cantilever is at the TOP (opposite to a simply-supported beam where tension is at the bottom).
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