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Unformatted text preview: Notes for Day : Â§ . : App ications to Mechanics You should be familiar by now with the â€śclassicalâ€ť model of a falling object in which: a ( t ) =- g v ( t ) =- gt + v s ( t ) =- gt + v + s , which is obtained by setting acceleration equal to a gravitational constant (such as . m s ; the negative sign indicates a downward direction) and integrating twice. Weâ€™re in a position now to consider more realistic models in which friction due to air resistance plays a role. We consider two cases: Drag Force Proportiona to Ve ocity In the rst case, the friction due to air resistance applies a drag force which slows down the object. e drag force acts on the object in the direction opposite to the objectâ€™s velocity (in other words, drag reduces speed). Newtonâ€™s Second Law of motion tells us that F = ma , or in other words, a = F m . Since a , acceleration, is the derivative of velocity, we can write dv dt = F m ere are two forces weâ€™re modelling: gravity, which is proportional to the objectâ€™s mass (i.e. gm ) and drag force, which is proportional to the objects velocity (i.e. kv ). e total force acting on the object, then is: F = - gm- kv , where the negative signs indicate the direction of the force. Combining this equation with the one above, we get: dv dt = - g- kv m is is a linear, nonhomogeneous di erential equation....
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This note was uploaded on 04/05/2008 for the course MATH 2214 taught by Professor Edesturler during the Spring '06 term at Virginia Tech.
- Spring '06