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# lab02.desc - LAB#2 Escape Velocity Goal Determine the...

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LAB #2 Escape Velocity Goal : Determine the initial velocity an object is shot upward from the surface of the earth so as to never return; illustrate scaling variables to simplify differential equations. Required tools : dfield and pplane ; second order differential equations; convert- ing a second order differential equation with missing independent variable to a first order differential equation; converting a second order differential equation to a system of differential equations. Discussion Newton’s Law of Gravitational Attraction tells us that in general an object of mass m which is x miles above the surface of the earth will feel a force of approximately F = - 0 . 0061 m 1 + x 4000 2 pounds with the number “4000” an approximation to the radius of the earth in miles and “0.0061” the acceleration due to gravity measured in miles/sec 2 . On the other hand, according to Newton’s 2 nd Law F = m d 2 x dt 2 . Thus equating these expressions for F leads to the differential equation d 2 x dt 2 = - 0 . 0061 1 + x 4000 2 . To study this equation using the Student Edition of Matlab , we need to change the units to avoid overly large intervals. Rather than measuring distance in miles, we will use multiples of the radius of the earth. This amounts to making the substitution y = x 4000 (scaling the dependent variable) resulting in the equation d 2 y dt 2 = - 0 . 0061 4000 1 (1 + y ) 2 . ( A ) We can further simplify this equation by changing the units of time (scaling the independent variable). If we let s = 0 . 0061 4000 t 1

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then by the Chain Rule dy dt = dy ds ds dt = 0 . 0061 4000 dy ds and hence d 2 y dt 2 = d dt dy dt = d dt 0 . 0061 4000 dy ds d 2 y dt 2 = 0 . 0061 4000 d dt dy ds d 2 y dt 2 = 0 . 0061 4000 d ds dy ds ds dt d 2 y dt 2 = 0 . 0061 4000 d 2 y ds 2 Using the last expression, the differential equation (A) becomes d 2 y ds 2 = - 1 (1 + y ) 2 . ( B ) Hence y is now viewed as a function of s
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lab02.desc - LAB#2 Escape Velocity Goal Determine the...

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