Question

Position of a body with acceleration \(\alpha\) is given by \(x = k \alpha^m t^n\), here \(t\) is time. Find the dimension of \(m\) and \(n\)

• A. \(m = 1, n = 1\)
• B. \(m = 1, n = 2\)
• C. \(m = 2, n = 1\)
• D. \(m = 2, n = 2\)

Hint:

For constant acceleration, \(x = \frac{1}{2}\alpha t^2\). So \(m = 1, n = 2\) matches.

Answer 1

The Correct Option is B

Step 1: Concept:
Dimensions: \([x] = L\), \([\alpha] = LT^{-2}\), \([t] = T\).
Step 2: Explanation:
\([x] = [\alpha]^m [t]^n \Rightarrow L = (LT^{-2})^m (T)^n = L^m T^{-2m + n}\).
Equating powers: For L: \(1 = m \Rightarrow m = 1\).
For T: \(0 = -2m + n = -2(1) + n \Rightarrow n = 2\).
Step 3: Conclusion:
Thus, \(m = 1, n = 2\).