question_answer

The SI unit of energy is\[J=kg{{m}^{2}}{{s}^{-2}},\]that of speed \[\upsilon \] is \[\mathbf{m}{{\mathbf{s}}^{-\mathbf{1}}}\]and acceleration a is \[\mathbf{m}{{\mathbf{s}}^{-\mathbf{2}}}\]. Which of the formulae for kinetic energy (K) given below can you rule out on the basis of dimensional arguments (m stands for the mass of the body) : (a) \[K={{m}^{2}}{{\upsilon }^{3}}\]       (b) \[K=\left( 1/2 \right)m{{\upsilon }^{2}}\] (c) \[~K=ma\]           (d) \[~K=\text{ }(\mathbf{3}\text{ }/16)m{{\upsilon }^{2}}\] (e) \[K=(\mathbf{1}/\text{ 2})m{{\upsilon }^{2}}+\text{ }ma\]

Answer:

                            As SI unit of energy, \[J=kg{{m}^{2}}{{s}^{-2}},\]so [Energy] \[=[M{{L}^{2}}{{T}^{-2}}]\] (a) \[[{{m}^{2}}{{\upsilon }^{2}}]=[{{M}^{2}}]{{[L{{T}^{-1}}]}^{2}}=[{{M}^{2}}{{L}^{2}}{{T}^{-2}}]\] (b) \[[1/2m{{\upsilon }^{2}}]=[M]{{[L{{T}^{-1}}]}^{2}}=[M{{L}^{2}}{{T}^{-2}}]\] (c) \[[ma]=[M][L{{T}^{-2}}]=[ML{{T}^{-2}}]\] (d) \[[3/16m{{\upsilon }^{2}}]=[M]{{[L{{T}^{-1}}]}^{2}}=[M{{L}^{2}}{{T}^{-2}}]\] (e) The quantities \[\left( \text{1}/\text{2} \right)m{{\upsilon }^{2}}\]and ma have different dimensions and hence cannot be added. Since the kinetic energy K has the dimensions of\[[M{{L}^{2}}{{T}^{-2}}],\] formulas (a), (c) and (e) are clearly ruled out. Dimensional analysis cannot tell which of the two, (b) or (d), is the correct formula. From the actual definition of kinetic energy, only (b) is the correct formula for kinetic energy.