Question

With the usual notations, the following equations $s_t=u+\frac{1}{2}a(2t-1)$ is

• A. Only numerically correct
• B. Only dimensionaly correct
• C. Both numerically and dimensionally correct
• D. Neither numerically nor dimensionally correct

Answer 1

Answer: (A)

Only numerically correct

Dimensions of LHS is: $\left[L\right]$

RHS has three terms:

$u$ has the dimensions of speed: $\left[L^1{T}^{-1}\right]$

$at$ has the dimensions: $\left[L^1{T}^{-2}\right]\left[T^1\right]=\left[L^1{T}^{-1}\right]$

$-a$ has the dimension of acceleration: $\left[L^1{T}^{-2}\right]$

Two quantities can be added only if they have the same dimensions.

Also, two quantities can be equal only if they have the same dimensions. Hence, the equation is dimensionally incorrect.

Correct equation is $s=ut+\frac{1}{2}a{t}^2$

Distance moved by the body in time-interval $(t-1,t)$ of 1 sec on solving comes out to be:

$s=u+\frac{a}{2}(2t-1)$

Hence, the given equation is numerically correct.

The equation can be made dimensionally correct by introducing new constants of unit magnitude.

$s_t=k_{1}u+\frac{k_{1}a}{2}(2t-k_1)$

$k_1=1s$

Hence, Correct answer is (a)