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

In the given expression, the term \( At^2 \) has the dimension of velocity, which is \( [L T^{-1}] \). To maintain dimensional consistency, the term \( A \) must have the dimensions: \[ A \sim \frac{[L T^{-1}]}{[T^2]} = [M L^2 T^{-3}] \] Similarly, the dimension of \( B \) and \( C \) can be derived to ensure the dimensions of the overall expression match those of velocity. The final result is \( [M L^2 T^{-3}] \). Thus, the correct answer is \( [M L^2 T^{-3}] \).