Suppose that an object is moving along a vertical line, and its vertical position is given by L (t). velocity of the object between t 1 and t 2 is

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1 Lecure Noes Insananeous Velociy age Suose a an objec is moving along a verical line, and is verical osiion is given by L (). velociy of e objec beween and 2 is Te average v av = L ( 2) L ( ) 2 We de ne e insanenous velociy a as e i of e average velociies, were e ime inerval around is geing smaller and smaller. In sor, e insananeous velociy a ime is e following i (if is i exiss) L ( + ) L () L ( + ) L () v () + Samle Problems. Te locaion funcion of an objec is L () = 2 3. Comue e insananeous velociy of e objec a) a = 7 second b) a = 0 second c) a. 2. Te locaion funcion of an objec is L () = 3. Comue e insananeous velociy of e objec a) a = 4 second b) a. 3. Te locaion funcion of an objec is L () =. Comue e insananeous velociy of e objec a) a = 49 second b) a 4. Te locaion funcion of an objec is L () =. Comue e insananeous velociy of e objec a) a = 5 second b) a Pracice Problems. Te locaion funcion of an objec is L () = 2 +. Comue e insananeous velociy of e objec a) a = 3 second b) a = 4 second c) a 2. Te locaion funcion of an objec is L () = 4. Comue e insananeous velociy of e objec a) a = 3 second b) a (Hin: you may need e following formula: (x + y) 4 = x 4 + 4x 3 y + 6x 2 y 2 + 4xy 3 + y 4 ) 3. Te locaion funcion of an objec is L () = 2 +. Comue e insananeous velociy of e objec a) a = 2 second b) a 4. Te locaion funcion of an objec is L () = a) a = 2 second b) a Comue e insananeous velociy of e objec 5. (Enricmen) Te locaion funcion of an objec is given by L () = Wen is e objec moving uward? c Hidegkui, 205 Las revised: Seember 5, 205

2 Lecure Noes Insananeous Velociy age 2 Answers - Samle Problems. a) v (7) = b) v (0) = 7 c) v () = a) v (4) = 48 b) v () = a) v (49) = 4 b) v () = 2 = 2 4. a) v (5) = 25 b) v () = 2 Answers - Pracice Problems. a) v (3) = 5 b) v (4) = 7 c) v () = a) v (3) = 08 b) v () = a) v (2) = 5 b) L 0 () = a) v (2) = 3 2 b) v () = 3 (3 + 5) 2 Samle Problems - Soluions. Te locaion funcion of an objec is L () = 2 3. a) Comue e insananeous velociy of e objec a = 7 second. We comue rs L (7 + ) v 7 L (7 + ) L (7) L (7 + ) = (7 + ) 2 3 (7 + ) = = We also comue L (7) So now e velociy: L (7) = = 49 2 = 38 L (7 + ) L (7) v 7 ( + ) = So a = 7, e velociy of e objec is. In sor, v (7) = : / ( + ) / c Hidegkui, 205 Las revised: Seember 5, 205

3 Lecure Noes Insananeous Velociy age 3 b) Comue e insananeous velociy of e objec a = 0 second. We comue rs L (0 + ) v (0) L (0 + ) L (0) L (0 + ) = (0 + ) 2 3 (0 + ) = = We also comue L (0) So now e velociy: L (0) = = = 70 L (0 + ) L (0) v (0) ( + 7) = 7 So a = 0, e velociy of e objec is 7. In sor, v (0) = / ( + 7) / c) Comue e insananeous velociy of e objec a. If we do a and we obain an exression in erms of ; en we creaed a new funcion, e velociy funcion. L ( + ) L () v () We comue rs L ( + ) L ( + ) = ( + ) 2 3 ( + ) = So now e velociy: L (0 + ) L (0) v (0) ( + 2 3) = / ( + 2 3) / So if an objec s locaion is given by L () = 2 3, en is velociy a ime is v () = 2 3. If we look a is formula, v (7) = = and v (0) = = 7 agrees wi revious ndings. c Hidegkui, 205 Las revised: Seember 5, 205

4 Lecure Noes Insananeous Velociy age 4 2. Te locaion funcion of an objec is L () = 3. a) Comue e insananeous velociy of e objec a = 4 second. We comue rs L (4 + ) v (4) L (4 + ) L (4) L (4 + ) = (4 + ) 3 = = We also comue L (4) = 64. So now e velociy: L (4 + ) L (4) / v (4) / = 48 So a = 4, e velociy of e objec is 48. In sor, v (4) = 48: b) Comue e insananeous velociy of e objec a. If we do a and we obain an exression in erms of ; en we creaed a new funcion, e velociy funcion. L ( + ) L () v () We comue rs L ( + ) So now e velociy: L ( + ) = ( + ) 3 = v () = L ( + ) L () = / / = So if an objec s locaion is given by L () = 3, en is velociy a ime is v () = 3 2. formula, v (4) = = 48 agrees wi revious ndings. If we look a is 3. Te locaion funcion of an objec is L () =. a) Comue e insananeous velociy of e objec a = 49 second. L (49 + ) L (49) v (49) Since is is an indeerminae wi radicals, we will use e conjugae of v (49) = = / / = So a = 49, e velociy of e objec is 4. In sor, v (49) = 4. c Hidegkui, 205 Las revised: Seember 5, 205

5 Lecure Noes Insananeous Velociy age 5 b) Comue e insananeous velociy of e objec a. L ( + ) L () v () + Since is is an indeerminae wi radicals, we will use e conjugae of v (49) / / + + = So if an objec s locaion is given by L () =, en is velociy a ime is v () = 2. If we look a is formula, v (49) = 2 49 = agrees wi revious ndings Te locaion funcion of an objec is L () =. a) Comue e insananeous velociy of e objec a = 5 second. L (5 + ) L (5) v (5) (5 + ) (5 + ) 5 (5 + ) 5 (5 + ) 5 (5 + ) = 25 So a = 5, e velociy of e objec is. In sor, v (5) =. Te negaive sign ere indicaes a e objec is moving downward a = 5 second. b) Comue e insananeous velociy of e objec a. L ( + ) L () v () ( + ) + ( + ) ( + ) ( + ) ( + ) = 2 So if an objec s locaion is given by L () =, en is velociy a ime is v () = formula, v (5) = agrees wi revious ndings. 25. If we look a is 2 For more documens like is, visi our age a :// and click on Lecure Noes. quesions or commens o midegkui@ccc.edu. c Hidegkui, 205 Las revised: Seember 5, 205