Which of the following equations are dimensionally consistent? (a) 𝒙 = 𝟏 𝟐 𝒂𝒕 𝟐 , (b) t=v/x, (c) t= (𝟐𝒙/𝒂) 𝟏/𝟐

Question

Which of the following equations are dimensionally consistent? (a) 𝒙 = 𝟏 𝟐 𝒂𝒕 𝟐 , (b) t=v/x, (c) t= (𝟐𝒙/𝒂) 𝟏/𝟐

Answer 1

Each of the equations you listed above consist of various combinations of the dimensions length (L) and time (T). To determine whether an equation is dimensionally consistent, begin by writing out the dimensions of each component on each side of the equation. For instance, in the first equation, x = 1/2 at², we have the following:

x = 1/2 a t²

L = L/T² T² (Note that no dimensions are associated with the 1/2)

Next, simplify the combinations of dimensions on each side of the equation. Here the left is already simplified. On the right, multiplying (L/T²) by T² results in L.

Finally, compare the left and right sides of the equation. Here, with L on each side, we have agreement, so the equation is dimensionally consistent.

For the second equation, we have T on the left and (L/T) / L on the right. Simplifying the right hand side gives us 1/T, which is not the same as the T on the left, so this equation is not consistent. (The correct expression here would instead be t = x/v.)

For the third equation, we have T on the left. On the right, we find the square root of L/ (L/T²). This expression reduces to the square root of T², which is consistent with the T on the left.