Question

The velocity of an object varies with time as v = At² + Bt + C. If units of v and t are expressed in SI, find the units of constants A, B and C.

Answer 1

Dear Student,

$$\begin{gathered} \mathrm{v}={\mathrm{At}}^2+\mathrm{Bt}+\mathrm{C} \\ \mathrm{where},\mathrm{v}=\mathrm{velocity} \\ \mathrm{t}=\mathrm{time} \\ \mathrm{Dimensional}\mathrm{formula}\mathrm{of}\mathrm{v}=\left[{\mathrm{M}}^0{\mathrm{L}}^1{\mathrm{T}}^{-1}\right] \\ \mathrm{Dimensional}\mathrm{formula}\mathrm{of}\mathrm{t}=\left[{\mathrm{M}}^0{\mathrm{L}}^0{\mathrm{T}}^1\right] \\ \mathrm{v}={\mathrm{At}}^2 \\ \mathrm{A}=\frac{\mathrm{v}}{{\mathrm{t}}^2} \\ \mathrm{A}=\frac{\left[{\mathrm{M}}^0{\mathrm{L}}^1{\mathrm{T}}^{-1}\right]}{{\left[{\mathrm{M}}^0{\mathrm{L}}^0{\mathrm{T}}^1\right]}^2} \\ \mathrm{A}=\frac{\left[{\mathrm{M}}^0{\mathrm{L}}^1{\mathrm{T}}^{-1}\right]}{\left[{\mathrm{M}}^0{\mathrm{L}}^0{\mathrm{T}}^2\right]} \\ \mathrm{A}=\left[{\mathrm{M}}^0{\mathrm{L}}^1{\mathrm{T}}^{-1-2}\right]=\left[{\mathrm{M}}^0{\mathrm{L}}^1{\mathrm{T}}^{-3}\right] \\ \mathrm{Also},\mathrm{v}=\mathrm{Bt} \\ \mathrm{B}=\frac{\mathrm{v}}{\mathrm{t}} \\ \mathrm{B}=\frac{\left[{\mathrm{M}}^0{\mathrm{L}}^1{\mathrm{T}}^{-1}\right]}{\left[{\mathrm{M}}^0{\mathrm{L}}^0{\mathrm{T}}^1\right]} \\ \mathrm{B}=\left[{\mathrm{M}}^0{\mathrm{L}}^1{\mathrm{T}}^{-1-1}\right]=\left[{\mathrm{M}}^0{\mathrm{L}}^1{\mathrm{T}}^{-2}\right] \\ \mathrm{Again}, \\ \mathrm{v}=\mathrm{C} \\ \mathrm{C}=\left[{\mathrm{M}}^0{\mathrm{L}}^1{\mathrm{T}}^{-1}\right] \end{gathered}$$