Solve x = At^3 + Bt

Question

(a) Assume the equation x = At^3 + Bt describes the motion of a particular object, with x having the demension of length and t having the demension of time. Determine the demensions of the constants A and B.

(b) Determine the dimensions of the derivetive dx/dt = 3At^2 + B

Answer 1

Let [T] denote the dimension time and [L] distance (length).

The dimension equation is:

[L]=A[T]³+B[T]. Therefore A=[L][T]^-3 and B=[L][T]^-1 as dimensional quantities.

dx/dt=3At²+B, which dimensionally is:

[L][T]^-1=[L][T]^-3[T]²+[L][T]^-1=[L][T]^-1, a balanced equation dimensionally.