Problem — Vertical projectile with drag

Problem

A $20\;\mathrm{g}$ projectile is fired vertically upward with initial speed $15\;\mathrm{m/s}$. Neglecting air resistance, how high does it go? If instead air resistance dissipates $30\%$ of the initial kinetic energy, how high does it go?

Solution

Ideal case (no drag). Mechanical energy is conserved: at the highest point the speed is zero and all the initial kinetic energy has become potential. $\frac{1}{2}\,m\,v_0^2 = m\,g\,h_1 \quad\Rightarrow\quad h_1 = \frac{v_0^2}{2g} = \frac{(15\;\mathrm{m/s})^2}{2\cdot 9{,}8\;\mathrm{m/s^2}} = \frac{225}{19{,}6}\;\mathrm{m}$ $\ev{h_1 \approx 11{,}5\;\mathrm{m}}$ (The mass cancels out, so the result does not depend on the $20\;\mathrm{g}$.)

Real case (with drag). Air resistance dissipates $30\%$ of the initial kinetic energy: only $70\%$ converts into potential energy. $m\,g\,h_2 = 0{,}70\cdot \frac{1}{2}\,m\,v_0^2 = 0{,}70\cdot m\,g\,h_1$ $h_2 = 0{,}70\cdot h_1 = 0{,}70\cdot 11{,}5\;\mathrm{m}$ $\ev{h_2 \approx 8{,}0\;\mathrm{m}}$

Collegamenti

Argomenti: Lavoro ed energia Concetti: Conservazione dell’energia meccanica · Energia termica da attrito Competenze: Conservazione dell’energia