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 dimensional formula of \( v \) is \( [L T^{-1}] \).
- The dimensional formula of \( \frac{At^2 + Bt}{C + t} \) must be \( [L T^{-1}] \), aligning with \( v \). 
- For \( At^2 + Bt \), both terms require consistent dimensions. 
- Given \( A \) has dimensions \( [ML^2T^{-3}] \), \( A t^2 \) results in \( [ML^2T^{-1}] \), which then matches the \( [L T^{-1}] \) dimension of velocity after division by \( t^2 \) in the denominator. 
- \( B \) has dimensions \( [MLT^{-3}] \) to balance the velocity dimension when multiplied by \( t \). 
- The denominator \( C + t \) has dimensions of \( [T] \). Therefore, \( C \) must possess dimensions of \( [L T^{-2}] \). 
Consequently, the dimensions of \( A \), \( B \), and \( C \) are \( [ML^2T^{-3}] \), \( [MLT^{-3}] \), and \( [L T^{-2}] \) respectively.