question 12 the position, x, of an object is given by the equation x = a + bt+ct², where t refers to time…

Explanation:

Step1: Recall dimension - homogeneity principle

In an equation, all terms must have the same dimensions. The dimension of position (x) is length (L).

Step2: Analyze the dimension of term (A)

Since (x = A+Bt + Ct^{2}), and (x) has dimension (L), for the term (A) to be added to (Bt) and (Ct^{2}), (A) must have the dimension of length. So, ([A]=L).

Step3: Analyze the dimension of term (Bt)

The dimension of (t) is time (T). Let the dimension of (B) be ([B]). Since (Bt) must have the dimension of length (to be added to (A) and (Ct^{2})), we have ([B][T]=L). Solving for ([B]), we get ([B]=\frac{L}{T}).

Step4: Analyze the dimension of term (Ct^{2})

Let the dimension of (C) be ([C]). Since (Ct^{2}) must have the dimension of length, and the dimension of (t^{2}) is (T^{2}), we have ([C][T^{2}]=L). Solving for ([C]), we get ([C]=\frac{L}{T^{2}}).

Answer:

([A]=L), ([B]=\frac{L}{T}), ([C]=\frac{L}{T^{2}})