Question

The equation s = ut + 1/2 at² relates the displacement s, initial velocity u, time t and acceleration a, of a particle under uniform accelerated motion. Show that the equation is homogeneous. Is it consistent and physically correct?

Answer 1

Explanation

• To check for homogeneity & physiclly correction, perform the dimensional analysis

$\qquad\qquad \begin{aligned} \small r&=p\\ \small LHS:[S]&= \small L\\ \small RHS: [ut]&= \small LT^{-1}\times T=L\\ \small [1/2at^2]&= \small LT^{-2}\times T^2=L\\ \small LHS&=\small RHS \end{aligned}$

• Therefore, the equation is homogeneous

• This is one of the 4 kinematic equations which are collectively manipulated (most of the time) to work out a motion problem.
• And those are strictly restrained to motions with a constant acceleration.
• Therefore, for a true/valid motion, we are given at least a set of solution which satisfies all those 4 equations. (for a projectile motion there is given 2 values for the time at a given height)
• Therefore, as long as the acceleration is constant this equation (along with the other 3) becomes consistent.