Question

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

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

Hint:

For dimension analysis, ensure that both the numerator and denominator of a formula have consistent dimensions that match the expected units of the result.

Answer 1

The Correct Option is A

- The dimension of \( v \) is \( [L T^{-1}] \). 
- The dimension of \( \frac{At^2 + Bt}{C + t} \) should match the dimension of \( v \), i.e., \( [L T^{-1}] \). 
- For the numerator \( At^2 + Bt \), dimensions of both terms must be consistent. 
- \( A \) has dimensions of \( [ML^2T^{-3}] \), as \( A t^2 \) gives \( [ML^2T^{-1}] \), which balances the \( [L T^{-1}] \) dimension of velocity. 
- \( B \) has dimensions of \( [MLT^{-3}] \), as it has to balance the dimension of velocity when multiplied by \( t \). 
- The denominator \( C + t \) has dimensions of \( [T] \), so \( C \) must have dimensions \( [L T^{-2}] \). 
Thus, the dimension of \( A \), \( B \), and \( C \) is \( [ML^2T^{-3}] \).