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Unformatted text preview: oldhomewk 05 – YOO, HEE – Due: Jan 27 2008, 4:00 am 1 Question 1, chap 2, sect 7. part 1 of 1 10 points An object is released from rest at time t = 0 and falls through the air, which exerts a resistive force such that the acceleration a of the object is given by a = g − b v , where v is the object’s speed and b is a constant. If limiting cases for large and small values of t are considered, which of the following is a possible expression for the speed of the object as an explicit function of time? 1. v = g ( 1 − e − bt ) b correct 2. v = ( g e bt ) b 3. v = g t − b t 2 4. v = ( g + a ) t b 5. v = v + g t, v negationslash = 0 Explanation: At time t = 0, the speed of the object is zero, and at time t = ∞ , the acceleration is zero, corresponding to a speed v = g b . Check the five choices, and it shows that the only possible answer is v = g ( 1 − e − bt ) b . Note: The answer can be directly obtained by integration: a = g − b v d v dt = − g parenleftbigg b v g − 1 parenrightbigg integraldisplay v dv b g v − 1 = − g integraldisplay t dt g b ln parenleftbigg b g v − 1 parenrightbiggvextendsingle vextendsingle vextendsingle vextendsingle v = − g t vextendsingle vextendsingle vextendsingle vextendsingle t g b bracketleftBig ln parenleftbigg b g v − 1 parenrightbigg − ln( − 1) bracketrightBig = − g t ln parenleftbigg 1 − b g v parenrightbigg = − b t 1 − b g v = e − b t b g v = 1 − e − b t v = g b parenleftBig 1 − e − b t parenrightBig ....
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