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
2. samarm91 reblogged this from sayitwithscience and added:
Thanks
3. invictascientia reblogged this from sayitwithscience
4. 32squared reblogged this from sayitwithscience and added:
I feel so smart for actually understanding this! Thank you Calc class.
5. emmelius reblogged this from sayitwithscience and added:
May the mass times acceleration be with you!!
6. clammytower39 reblogged this from sayitwithscience
7. drpepper-queen reblogged this from sayitwithscience and added:
∑ F = ma … is a differential equation: where acceleration a(t), velocity v(t), and displacement s(t) are all vectors and...
8. takemetobroadway reblogged this from sayitwithscience
9. dontpanicitsjustscience reblogged this from likeaphysicist
10. the11dimensions reblogged this from likeaphysicist and added:
… is a differential
11. sciencemicroblog reblogged this from likeaphysicist
12. supereffectivechrisg reblogged this from likeaphysicist
13. likeaphysicist reblogged this from sayitwithscience
14. tryingtolovemyfuture reblogged this from sayitwithscience
15. danijatomi reblogged this from sayitwithscience
16. chibisokka reblogged this from biognosis
17. sayitwithscience posted this