A physical quantity Q is defined by the relation: Q = t^2 + x + v where t represents time, x represents distance, and v represents velocity. The dimensions of the constants , , and are given as [ ] = [ M L ^2 T ⁻⁴] , [ ] = [ L ] , and [ ] = [ M T ⁻¹] . Identify the physical quantity Q .

Step-by-step solution

According to the principle of dimensional homogeneity, each term in an equation must have the same dimensions. Let us find the dimension of the first term, t^2 + x . The dimension of the numerator is [ ][t^2] = [ M L ^2 T ⁻⁴][ T ^2] = [ M L ^2 T ⁻²] . Since x is distance, [ + x] = [ L ] . Thus, the dimension of the first term is [ M L ^2 T ⁻²] [ L ] = [ M L T ⁻²] . Now, let us verify with the second term, v . The dimension is [ ][v] = [ M T ⁻¹][ L T ⁻¹] = [ M L T ⁻²] . Since both terms have the dimension [ M L T ⁻²