Equations of linear motion for uniform acceleration, Physics

Given a particle moving along a straight line with fixed acceleration a. Let u be the initial velocity of the particle and v be the velocity at time t. So by definition,

                                                                              2065_linear motion.png               dv = a dt

                        Integrating, 479_linear motion1.png

(a being a constant may be taken out of the integral)                 v - u = at

                                           v = u + at                                                                     ...(i)

 

Further,  2361_linear motion2.png           dx = vdt

 

But,      v = u + at,               dx = (u + at)dt

 

Integrating within limits, equations change into,  949_linear motion3.png               

 

2249_linear motion4.png

                                                                                                          ...(ii)

Again, going back to definition of acceleration, we have,    306_linear motion5.png

Multiply and divide RHS of above equation by dx, equation gets,   406_linear motion6.png 

                                      1540_linear motion7.png      adv = adx

Integrating within limits, equation converts,      1709_linear motion8.png            →    291_linear motion9.png   

1370_linear motion10.png      →          1166_linear motion11.png

                                                                       ...(iii)

From (i) and (iii) we also result that,   2162_linear motion12.png                                       ...(iv)

Equation (iv) is a special equation, as we do not have a in this equation, but still it belongs to motion, which is uniformly accelerated.

To result, we again write the equation of motion for constant acceleration below.

v = u + at ;     s = ut + 557_linear motion13.png1395_linear motion14.png       695_linear motion15.png      

Posted Date: 8/3/2012 3:19:14 AM | Location : United States







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