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lecture_3 (dragged) - MA 36600 LECTURE NOTES FRIDAY JANUARY...

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MA 36600 LECTURE NOTES: FRIDAY, JANUARY 16 Solutions of Some Differential Equations Gravity without Air Resistance. Say that we have a mass m at a position x = x ( t ). We wish to discuss its motion under the influence of gravity. We assume for simplicity that the object is near sea level, at the surface of the Earth. #1: Our variables are time t and position x . Note that t is an independent variable, and x = x ( t ) is a dependent variable. #2: The principle that underlies the problem is Newton’s Law of Gravity i.e. F = m g . (This is only true for a mass that is near sea level!) #3: Our initial conditions are x (0) = x 0 and v (0) = v 0 . The initial value problem is m d 2 x dt 2 = m g Di ff erential Equation x (0) = x 0 v (0) = v 0 Initial Conditions We will show that the solution is x ( t ) = x 0 + v 0 t 1 2 g t 2 . To this end, first we consider the velocity v = x ( t ). We have the initial value problem dv dt = g Di ff erential Equation v (0) = v 0 Initial Condition The di ff erential equation is dv dt = g = v ( t ) = g t + C for some constant C . (You can see this by considering
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