Question

Which of the following is dimensionally incorrect?
(A) $ u = v - at $
(B) $ s - ut = \dfrac{1}{2}a{t^2} $
(C) $ {u^2} = 2a(gt - 1) $
(D) $ {v^2} - {u^2} = 2as $

Answer 1

Hint : When two quantities are being added together or subtracted from each other the unit, thus dimension must be the same. The dimension of the right hand side must be equal to the dimension on the left and side of an equation.

Complete step by step answer
To get the dimensionally incorrect formula, we investigate them one after the other based on these two principles;
1. Quantities that are being added to each other or subtracted from each other must have the same unit and dimension.
2. The dimension on the left hand side of an equation must be equal to the dimension on the right hand side of the equation.
Now, for option A, we have
 $ u = v - at $ , $ u $ is initial velocity, $ v $ is final velocity, $ a $ is acceleration and $ t $ is time.
Checking their dimension, we have
 $ L{T^{ - 1}} = L{T^{ - 1}} - \left( {L{T^{ - 2}} \times T} \right) $
 $ \Rightarrow L{T^{ - 1}} = L{T^{ - 1}} - \left( {L{T^{ - 1}}} \right) $ . Hence it is dimensionally correct.
Now, for option B, we have that
 $ s - ut = \dfrac{1}{2}a{t^2} $ where $ s $ is the distance
Hence, checking the dimension, we have
 $ L - \left( {L{T^{ - 1}} \times T} \right) = \left( {L{T^{ - 2}} \times {T^2}} \right) $
 $ \Rightarrow L - \left( L \right) = \left( L \right) $
Now, for option C, we have
 $ {u^2} = 2a(gt - 1) $ where $ g $ must be the acceleration due to gravity.
Hence, checking the dimension we have
 $ {\left( {L{T^{ - 1}}} \right)^2} = \left( {L{T^{ - 2}}} \right)\left( {\left( {L{T^{ - 2}} \times T} \right) - \left( {{L^0}{T^0}} \right)} \right) $
Now, from the first principle stated above, option C must be dimensionally incorrect because it has the subtraction of dimensionless constant with a quantity with dimension.
Hence, the correct option is option C.

Note
Alternatively, observing the equations in the options, we can see that option A, B, and D are all equations of motion, rearranged in one way or another. Hence, without actually checking the dimension, we can conclude that option C has an incorrect dimension. Thus, the correct answer is option C.