Question

The expression given below shows the variation of velocity \( v \) with time \( t \): \[ v = At^2 + \frac{Bt}{C + t}. \] The dimension of \( ABC \) is:

• A. \( [M L^2 T^{-3}] \)
• B. \( [M L T^{-3}] \)
• C. \( [M L^2 T^{-2}] \)
• D. \( [M L T^{-2}] \)

Hint:

In dimensional analysis, balance the dimensions on both sides of the equation. Each term must have the appropriate dimensions for consistency.

Answer 1

The Correct Option is A

For dimensional consistency in the expression, where the term \( At^2 \) has the dimension of velocity \( [L T^{-1}] \), the dimensions of \( A \) must be: \[A \sim \frac{[L T^{-1}]}{[T^2]} = [M L^2 T^{-3}]\ \] The dimensions of \( B \) and \( C \) are similarly determined to ensure the entire expression's dimensions align with velocity. The ultimate outcome is \( [M L^2 T^{-3}] \). Therefore, the correct answer is \( [M L^2 T^{-3}] \).