Physics Homework Help With Motion With Constant Acceleration

Motion With Constant Acceleration

In this section, we are going to derive relations among displacement, velocity, acceleration and time.
We know that,            
                             dv = adt

On integrating both sides,we get
         òdv = òadt
         v = at + c1           where c1 = constant of integration

If  v0 be the initial velocity of the particle then at t = 0, v = v0. This implies that c1 = v0
         v = v0 + at                                               … (1)
Also we know that
        
or      dx = (v0 + at)dt

On integrating, we get
         òdx = òv0dt + òatdt
         x = v0t + at2 + c2          where c2 = constant

If x0 be the initial position of the particle, then at t = 0; x = x0. This implies that c2 = x0
         x = x0 + v0t +  at2                     … (2)

Eliminating t from equations (1) and (2), we get
         v2 =                  … (3)

The equation (2) may be rewritten as
                 

From equation (1), we have
        
   
thus                         … (4)

The equations (1) to (4) are called the equations of kinematics  in the direction along the x-axis.
The equations of kinematics are summarized as
         v = v0 + at                                               (1)
         x = x0 + v0t +                       (2)
         v2 =                   (3)
         x = x0 + (v0 + v)t                     (4)

where x0 = initial position coordinate; x = Final position coordinate
           v = final velocity; a = acceleration (constant)

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