Difference between revisions of "Forest UCM PnCP LinAirRes"

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::<math>= \frac{v_0}{-b}\left ( e^{-bt}-e^{-b0}  \right ) + v_t t + \frac{v_t}{b}\left ( e^{-bt} - e^{-b0}\right )  dt</math>
 
::<math>= \frac{v_0}{-b}\left ( e^{-bt}-e^{-b0}  \right ) + v_t t + \frac{v_t}{b}\left ( e^{-bt} - e^{-b0}\right )  dt</math>
 
::<math>= \frac{v_0}{b}\left ( 1- e^{-bt}  \right ) + v_t t + \frac{v_t}{b}\left ( e^{-bt} - 1}\right )  dt</math>
 
::<math>= \frac{v_0}{b}\left ( 1- e^{-bt}  \right ) + v_t t + \frac{v_t}{b}\left ( e^{-bt} - 1}\right )  dt</math>
::<math>= v_t t \frac{1}{b}\left ( v_0 - v_t) \left ( 1- e^{-bt}  \right )  dt</math>
+
::<math>= v_t t \frac{1}{b}\left ( v_0 - v_t) \right ) \left ( 1- e^{-bt}  \right )  dt</math>
  
  
 
[[Forest_UCM_PnCP#Linear_Air_Resistance]]
 
[[Forest_UCM_PnCP#Linear_Air_Resistance]]

Revision as of 12:09, 1 September 2014

Linear Air Resistance

Horizontal motion

If n is unity then the velocity is exponentially approaching zero.

F(v)=bv: negative sign indicates a retarding force and b is a proportionality constant
Fext=bv=mdvdt
vfvidvv=tftibmdt
lnvfvi=bmt; ti0
vf=viebmt

The displacement is given by

x=t0viebmtdt
=vi(ebmtbm)|t0
=vi(mbebmt)|t0
=vi(mbebmt)|0t
=vi(mbebm0mbebmt)
=mbvi(1ebmt)


Example: falling object with linear air friction

Consider a ball falling under the influence of gravity and a frictional force that is proportion to its velocity

Fext=mgbv=mdvdt

let

b=coefficient of air resistance
vt=mgb= Terminal speed
vtv=1bdvdt
bdt=dvvtv
bdt=dvvvt
t0bdt=vv0dvvvt
bt=ln(vvt)ln(v0vt)
bt=ln(vvtv0vt)
ebt=(vvtv0vt)
vvt=(v0vt)ebt
v=v0ebt+vt(1ebt)


The posiiton as a function of time may be determined by directly integrating the above equation

dydt=v0ebt+vt(1ebt)
y0=t0(v0ebt+vt(1ebt))dt
y=t0v0ebtdt+t0vt(1ebt)dt
=v0b(ebteb0)+vtt+vtb(ebteb0)dt
= \frac{v_0}{b}\left ( 1- e^{-bt}  \right ) + v_t t + \frac{v_t}{b}\left ( e^{-bt} - 1}\right )  dt
=vtt1b(v0vt))(1ebt)dt


Forest_UCM_PnCP#Linear_Air_Resistance