Forest UCM PnCP LinAirRes - Revision history

Foretony: /* Example: falling object with linear air friction */

2014-09-01T12:43:14Z

Example: falling object with linear air friction

Foretony: /* Example: falling object with linear air friction */

2014-09-01T12:41:26Z

Example: falling object with linear air friction

Foretony: /* Example: falling object with linear air friction */

2014-09-01T12:17:30Z

Example: falling object with linear air friction

Foretony: /* Example: falling object with linear air friction */

2014-09-01T12:10:29Z

Example: falling object with linear air friction

Foretony: /* Example: falling object with linear air friction */

2014-09-01T12:09:51Z

Example: falling object with linear air friction

Foretony: /* Example: falling object with linear air friction */

2014-09-01T12:09:35Z

Example: falling object with linear air friction

Foretony: /* Example: falling object with linear air friction */

2014-09-01T12:02:31Z

Example: falling object with linear air friction

Foretony: /* Example: falling object with linear air friction */

2014-09-01T11:59:33Z

Example: falling object with linear air friction

Foretony: /* Example: falling object with linear air friction */

2014-09-01T11:56:37Z

Example: falling object with linear air friction

Foretony: /* Example: falling object with linear air friction */

2014-08-31T14:00:37Z

Example: falling object with linear air friction

@@ Line 36: / Line 36: @@
 :<math>  -\frac{b}{m}t = \ln{\left( v -v_t \right)} - \ln{\left ( v_0-v_t \right )}</math>
 :<math>  -\frac{b}{m}t = \ln \left(\frac{ v -v_t  }{v_0-v_t}\right )</math>
-:<math>  e^{-bt} =  \left(\frac{ v -v_t  }{v_0-v_t}\right )</math>
+:<math>  e^{-\frac{b}{m}t} =  \left(\frac{ v -v_t  }{v_0-v_t}\right )</math>
 :<math>  v -v_t = \left ( v_0-v_t\right )e^{-\frac{b}{m}t}</math>
 :<math>  v  = v_0e^{-\frac{b}{m}t} + v_t \left (1 -e^{-\frac{b}{m}t}\right )</math>
@@ Line 46: / Line 46: @@
 :<math> \int_0^y = \int_0^t \left (  v_0e^{-\frac{b}{m}t} + v_t \left (1 -e^{-\frac{b}{m}t}\right ) \right ) dt</math>
 :<math>y = \int_0^t v_0e^{-\frac{b}{m}t}dt  + \int_0^t v_t \left (1 -e^{-b\frac{b}{m}t}\right )  dt</math>
-::<math>= \frac{v_0}{-\frac{b}{m}}\left ( e^{-\frac{b}{m}t}-e^{-b0}  \right ) + v_t t + \frac{mv_t}{b}\left ( e^{-\frac{b}{m}t} - e^{-b0}\right )  </math>
+::<math>= \frac{v_0}{-\frac{b}{m}}\left ( e^{-\frac{b}{m}t}-e^{-b0}  \right ) + v_t t + \frac{mv_t}{b}\left ( e^{-\frac{b}{m}t} - e^{-\frac{b}{m}0}\right )  </math>
 ::<math>= \frac{v_0}{\frac{b}{m}}\left ( 1- e^{-\frac{b}{m}t}  \right ) + v_t t + \frac{v_t}{b}\left ( e^{-bt} - 1\right )  </math>
 ::<math>= v_t t + \frac{m}{b}\left ( v_0 - v_t) \right ) \left ( 1- e^{-\frac{b}{m}t}  \right )  </math>

@@ Line 30: / Line 30: @@
 :<math>b=</math>coefficient of air resistance
 :<math>v_t = \frac{mg}{b} =</math> Terminal speed
-:<math> v_t -v = \frac{1}{b} \frac{dv}{dt}</math>
+:<math> v_t -v = \frac{m}{b} \frac{dv}{dt}</math>
-:<math>  b dt= \frac{dv}{v_t -v} </math>
+:<math>  \frac{b}{m} dt= \frac{dv}{v_t -v} </math>
-:<math>  -b dt= \frac{dv}{v -v_t} </math>
+:<math>  -\frac{b}{m} dt= \frac{dv}{v -v_t} </math>
-:<math>  -\int_0^t b dt= \int_{v_0}^v \frac{dv}{v -v_t} </math>
+:<math>  -\int_0^t \frac{b}{m} dt= \int_{v_0}^v \frac{dv}{v -v_t} </math>
-:<math>  -bt = \ln{\left( v -v_t \right)} - \ln{\left ( v_0-v_t \right )}</math>
+:<math>  -\frac{b}{m}t = \ln{\left( v -v_t \right)} - \ln{\left ( v_0-v_t \right )}</math>
-:<math>  -bt = \ln \left(\frac{ v -v_t  }{v_0-v_t}\right )</math>
+:<math>  -\frac{b}{m}t = \ln \left(\frac{ v -v_t  }{v_0-v_t}\right )</math>
 :<math>  e^{-bt} =  \left(\frac{ v -v_t  }{v_0-v_t}\right )</math>
-:<math>  v -v_t = \left ( v_0-v_t\right )e^{-bt}</math>
+:<math>  v -v_t = \left ( v_0-v_t\right )e^{-\frac{b}{m}t}</math>
-:<math>  v  = v_0e^{-bt} + v_t \left (1 -e^{-bt}\right )</math>
+:<math>  v  = v_0e^{-\frac{b}{m}t} + v_t \left (1 -e^{-\frac{b}{m}t}\right )</math>
 The posiiton as a function of time may be determined by directly integrating the above equation
-:<math> \frac{dy}{dt} = v_0e^{-bt} + v_t \left (1 -e^{-bt}\right )</math>
+:<math> \frac{dy}{dt} = v_0e^{-bt} + v_t \left (1 -e^{-\frac{b}{m}t}\right )</math>
-:<math> \int_0^y = \int_0^t \left (  v_0e^{-bt} + v_t \left (1 -e^{-bt}\right ) \right ) dt</math>
+:<math> \int_0^y = \int_0^t \left (  v_0e^{-\frac{b}{m}t} + v_t \left (1 -e^{-\frac{b}{m}t}\right ) \right ) dt</math>
-:<math>y = \int_0^t v_0e^{-bt}dt  + \int_0^t v_t \left (1 -e^{-bt}\right )  dt</math>
+:<math>y = \int_0^t v_0e^{-\frac{b}{m}t}dt  + \int_0^t v_t \left (1 -e^{-b\frac{b}{m}t}\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 )  </math>
+::<math>= \frac{v_0}{-\frac{b}{m}}\left ( e^{-\frac{b}{m}t}-e^{-b0}  \right ) + v_t t + \frac{mv_t}{b}\left ( e^{-\frac{b}{m}t} - e^{-b0}\right )  </math>
-::<math>= \frac{v_0}{b}\left ( 1- e^{-bt}  \right ) + v_t t + \frac{v_t}{b}\left ( e^{-bt} - 1\right )  </math>
+::<math>= \frac{v_0}{\frac{b}{m}}\left ( 1- e^{-\frac{b}{m}t}  \right ) + v_t t + \frac{v_t}{b}\left ( e^{-bt} - 1\right )  </math>
-::<math>= v_t t + \frac{1}{b}\left ( v_0 - v_t) \right ) \left ( 1- e^{-bt}  \right )  </math>
+::<math>= v_t t + \frac{m}{b}\left ( v_0 - v_t) \right ) \left ( 1- e^{-\frac{b}{m}t}  \right )  </math>
 [[Forest_UCM_PnCP#Linear_Air_Resistance]]

@@ Line 46: / Line 46: @@
 :<math> \int_0^y = \int_0^t \left (  v_0e^{-bt} + v_t \left (1 -e^{-bt}\right ) \right ) dt</math>
 :<math>y = \int_0^t v_0e^{-bt}dt  + \int_0^t v_t \left (1 -e^{-bt}\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 ( e^{-bt}-e^{-b0}  \right ) + v_t t + \frac{v_t}{b}\left ( e^{-bt} - e^{-b0}\right )  </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 )  </math>
-::<math>= v_t t + \frac{1}{b}\left ( v_0 - v_t) \right ) \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 )  </math>
 [[Forest_UCM_PnCP#Linear_Air_Resistance]]

@@ Line 47: / Line 47: @@
 :<math>y = \int_0^t v_0e^{-bt}dt  + \int_0^t v_t \left (1 -e^{-bt}\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) \right ) \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]]

@@ Line 48: / Line 48: @@
 ::<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>= 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]]