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since you cannot simply use an equation like vf=vi+at, unless "a" is constant, how do you do it?

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- Thread starter an emu
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- #1

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since you cannot simply use an equation like vf=vi+at, unless "a" is constant, how do you do it?

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vanesch

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since you cannot simply use an equation like vf=vi+at, unless "a" is constant, how do you do it?

It's called "calculus" and was indeed the mathematical problem that Newton needed to solve before he could formulate his mechanics.

velocity is the integral of the acceleration.

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jtbell

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In planetary motion, how do you find change in velocity and change in time in relation to a change in distance? (a=GM/x^2)

Since planetary motion is two-dimensional (actually three-dimensional, but the orbit is confined to a plane if we're dealing with only one planet at a time, and neglecting perturbations from the other planets), you have to solve a pair of coupled differential equations:

[tex]\frac{d^2 x}{dt^2} = \frac{GMx}{(x^2 + y^2)^{3/2}}[/tex]

[tex]\frac{d^2 y}{dt^2} = \frac{GMy}{(x^2 + y^2)^{3/2}}[/tex]

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jtbell

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When you solve differential equations, you get x(t) and y(t), that is, formulas (or tables) for x and y at time t. The first derivatives dx/dt and dy/dt give you the x and y components of the velocity at time t. The second derivatives [itex]d^2 x / dt^2[/itex] and [itex]d^2 y / dt^2[/itex] give you the x and y components of the acceleration.

In practice, people usually solve differential equations like this using computer software. You give it the initial values of x and y at t = 0, and it calculates a table of x and y at later times. It calculates each point based on the results for the preceding point, going one step at a time. There are various methods (algorithms) for doing the calculation, with different combinations of simplicity, speed and accuracy: Euler's method, Runge-Kutta methods, etc. You typically learn the details in a numerical-methods course.

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