- Horizontal motion
If [math]n[/math] is unity then the velocity is exponentially approaching zero.
- [math]F(v) = -bv[/math]: negative sign indicates a retarding force and [math]b[/math] is a proportionality constant
- [math]\sum \vec {F}_{ext} = -bv = m \frac{dv}{dt}[/math]
- [math]\Rightarrow \int_{v_i}^{v_f} \frac{dv}{v} = \int_{t_i}^{t_f} \frac{-b}{m}dt[/math]
- [math]\ln\frac{v_f}{v_i} = \frac{-b}{m}t[/math]; [math]t_i \equiv 0[/math]
- [math]v_f = v_i e^{-\frac{b}{m}t}[/math]
The displacement is given by
- [math]x = \int_0^t v_i e^{-\frac{b}{m}t} dt[/math]
- [math]= \left . v_i \left ( \frac {e^{-\frac{b}{m}t}}{-\frac{b}{m}} \right ) \right |_0^t[/math]
- [math]= \left . v_i \left ( -\frac{m}{b} e^{-\frac{b}{m}t} \right ) \right |_0^t[/math]
- [math]= \left . v_i \left ( \frac{m}{b} e^{-\frac{b}{m}t} \right ) \right |_t^0[/math]
- [math]= v_i \left ( \frac{m}{b} e^{-\frac{b}{m}0} -\frac{m}{b} e^{-\frac{b}{m}t} \right ) [/math]
- [math]= \frac{m}{b} v_i \left ( 1-e^{-\frac{b}{m}t} \right )[/math]