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old 5 answer - oldhomewk 05 NOORANI ZOHEB Due 4:00 am...

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oldhomewk 05 – NOORANI, ZOHEB – 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 t b t 2 2. v = g ( 1 e bt ) b correct 3. v = ( g e bt ) b 4. v = ( g + a ) t b 5. v = v 0 + g t, v 0 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 0 dv b g v 1 = g integraldisplay t 0 dt g b ln parenleftbigg b g v 1 parenrightbigg vextendsingle vextendsingle vextendsingle vextendsingle v 0 = g t vextendsingle vextendsingle vextendsingle vextendsingle t 0 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 bt b g v = 1 e bt v = g b parenleftBig 1 e bt parenrightBig .
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