Tuesday, October 18, 2011

∑ F = ma

… is a differential equation:

where acceleration a(t), velocity v(t), and displacement s(t) are all vectors and functions of time. This equation is second-order in position because the highest derivative is the second time derivative of position. Combined with the right boundary conditions, s(t) (also called the trajectory: path through space and time) can be determined.

This differential equation can be solved one component, or dimension, at a time. Let us focus on one of these, and call it the x component. The equations for y and z can be found exactly the same way.

Constant acceleration

If the graph of a(t) signifying acceleration in the x direction is constant

then the graph of v(t), the velocity in the x direction, is a straight line with slope a0

and the graph of x(t), the position along the x axis, is a parabola

It is also possible for the acceleration, or either of the initial velocity or initial position, to be negative. Thus the displacement/projectile motion formula is derived.

Notes

  1. thestruggleofanamericanwriter reblogged this from sayitwithscience
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    Thanks
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    I feel so smart for actually understanding this! Thank you Calc class.
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    May the mass times acceleration be with you!!
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    ∑ F = ma … is a differential equation: where acceleration a(t), velocity v(t), and displacement s(t) are all vectors and...
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    … is a differential
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