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lecture_3 - MA 36600 LECTURE NOTES: FRIDAY, JANUARY 16...

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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 = - mg . (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 = - mg ± Differential 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 0 ( t ). We have the initial value problem dv dt = - g ± Differential Equation v (0) = v 0 ± Initial Condition The differential equation is dv dt = - g = v ( t ) = - g t + C for some constant C . (You can see this by considering V = v ( t ) + g t . Since V 0 ( t ) = 0, we see that V ( t ) = C must be a constant) Using the initial condition, we see that C = v (0) = v 0 . Hence the solution to the initial value problem is v ( t ) = v 0 - g t . We use this to find the solution to the original initial value problem. We have dx dt = v ( t ) = v 0 - g t ± Differential Equation x (0) = x 0 ± Initial Condition The differential equation is dx dt = v 0 - g t = x ( t ) = v 0 t - 1 2 g t 2 + C for some constant C . (As before, you can see this by considering X = x ( t ) - v 0 t + 1 2 g t 2 . Since X 0 ( t ) = 0 we see that X ( t ) = C must be a constant.) Using the initial condition, we see that C = x (0) = x 0 . Hence the solution to the original initial value problem is
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This note was uploaded on 05/24/2011 for the course MA 36600 taught by Professor Ma during the Spring '09 term at Purdue.

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lecture_3 - MA 36600 LECTURE NOTES: FRIDAY, JANUARY 16...

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