Question

If \[x = at + b{t^2}\]where x is in meters and t is in seconds. What are the units of a and b?

Answer 1

Hint:Units are the standards for the measurement of physical quantities that need clear definitions to be used. A unit of measurement is a definite magnitude of a quantity that is used as a standard for measurement of the same kind of quantity.
The dimension of a physical quantity is the power to which that fundamental unit is raised to obtain one unit of that quantity,To find the dimensions and units of more complex units, use the principle of dimensional homogeneity. In any valid physical equation, the dimensions of both sides of the physical equation must be equal.

Complete step-by-step solution
Here, in this question we need to determine the units for a and b such that \[x = at + b{t^2}\] should be satisfied with x in meters and t in seconds for which we need to compare both the sides of the equation with reference to the units of the variables.
Given equation is \[x = at + b{t^2}\], where the unit of x is in meters.
The equation shows x is the sum of \[at\] and \[b{t^2}\], hence the units of the individual terms also need to be in meter by the law of homogeneity,
 \[at = x \Rightarrow a = \dfrac{x}{t} = \dfrac{{\left[ L \right]}}{{\left[ T \right]}} = \left[ {L{T^{ - 1}}} \right] = \dfrac{m}{s}\]
\[b{t^2} = x \Rightarrow b = \dfrac{x}{{{t^2}}} = \dfrac{{\left[ L \right]}}{{\left[ {{T^2}} \right]}} = \left[ {L{T^{ - 2}}} \right] = \dfrac{m}{{{s^2}}}\]
Hence, the unit of a is \[meter/\sec \]and the unit of b is \[meter/\sec \] .

Note:It is always to be kept in mind that while writing the dimensional formula only and only SI units of the measuring quantities should be used and should be bifurcated further.