_finitedifexplan_pdf_ - Numerically integrating equations...

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Numerically integrating equations of motion Newtons equations give an equation for the acceleration of an object. ~a = ~ F m If you happen to know ~ F as a function of time, one can Fnd position as a function of time by integrating twice: ~v ( t )= ~v (0) + ! t 0 ~ F ( ¯ t ) m d ¯ t ~r ( t ~r (0) + ! t 0 ~v ( ¯ t ) d ¯ t. These equations just mean that you draw the graph of each component of ( t ) and then Fnd the areas to get ~v ( t ). You then repeat to get ~r ( t ). Life is usually more complicated. In most real-world applications the force is not given to you as a function of t , but instead the force depends on where the object is [or even how fast its going]. ±or example, specializing to one dimension, an object attached to a spring will experience a force F = kx, giving an equation of motion d 2 x ( t ) dt 2 = k m x ( t ) . Since this equation involves a derivative it is known as a “di²erential equa- tion”. It turns out that this simple example can be solved exactly, however at
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This note was uploaded on 03/27/2008 for the course PHYS 1112 taught by Professor Leclair,a during the Spring '07 term at Cornell University (Engineering School).

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_finitedifexplan_pdf_ - Numerically integrating equations...

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