Check the validity of the equation v = u + at^2 by its unit wise analysis.

Unit Analysis of $S=ut+\frac{1}{2}a{t}^2$

To check the validity of the equation, we compare the dimensions of the Left Hand Side (LHS) and Right Hand Side (RHS).

1. Left Hand Side (LHS): The term is $S$ (Displacement). $$\text{Dimension of }S=[L]$$

2. Right Hand Side (RHS): The term is $ut+\frac{1}{2}a{t}^2$.

• For $ut$:

  • $u$ (Initial velocity) has dimension $[L{T}^{-1}]$
  • $t$ (Time) has dimension $[T]$
  • $[ut]=[L{T}^{-1}]\times [T]=[L]$

• For $\frac{1}{2}a{t}^2$:

  • $a$ (Acceleration) has dimension $[L{T}^{-2}]$
  • $t^2$ has dimension $[T^2]$
  • $[\frac{1}{2}a{t}^2]=[L{T}^{-2}]\times [T^2]=[L]$

Conclusion: Since the dimension of LHS $[L]$ is equal to the dimension of RHS $[L]$, the equation is dimensionally correct (valid).