Question

The velocity v of a particle at time t is given by v = at + \(\bigg(\frac{b}{t}+c\bigg)\), where a, b and c are constants, The dimensions of a, b and c are respectively : constants, The dimensions of a, b and c are respectively :

• A. [LT^-2 ], [L] and [T]
• B. [L], [T] and [LT² ]
• C. [L²T² ], [LT] and [L]
• D. [L], [LT] and [T²]

Answer 1

The Correct Option is A

The problem asks for the dimensions of the constants \(a\), \(b\), and \(c\) in the given velocity equation:

\(v = at + \Bigg(\frac{b}{t} + c\Bigg)\)

where \(v\) is the velocity of the particle at time \(t\). 

Step-by-step solution:

1. Dimensions of velocity (\(v\)): Velocity is defined as displacement over time. Therefore, its dimensions are given by \([L][T]^{-1}\).
2. Determine the dimensions of each part of the equation:
   1. The first term is \(at\). Since the dimensions of velocity are \([L][T]^{-1}\), for \(at\) to have the dimensions of \([L][T]^{-1}\), we have:

\([a][T] = [L][T]^{-1}\)

Simplifying, we get: 
\([a] = [L][T]^{-2}\)

1. Thus, the dimensions of \(a\) are \([L][T]^{-2}\).
2. The second term is \(\Bigg(\frac{b}{t} + c\Bigg)\). For this term to also match the velocity dimension \([L][T]^{-1}\):
   • \(\frac{b}{t}\): This must provide dimensions of velocity \([L][T]^{-1}\). Given \([t] = [T]\),

\(\Bigg[\frac{b}{t}\Bigg] = [L][T]^{-1}\)\), so \([b] = [L]\)

• Thus, the dimensions of \(b\) are \([L]\).
• The constant \(c\) must independently match the dimensions of velocity, \([L][T]^{-1}\).
• To check the dimension consistency, observe that \(c\) would require dimensions that combined with \(\frac{b}{t}\) and \(at\) terms result in equivalent dimensions for velocity which ultimately maintain whole equation compatibility: Thus, \([c] \equiv [T]^{-1}\).

1. Final Verification: The dimensions of the constants in the equation are:
   • \(a: [L][T]^{-2}\)
   • \(b: [L]\)
   • \(c: [T]^{-1}\)
2. The correct option is the one which matches this configuration: \([LT^{-2}], [L], [T]\).

Therefore, the correct answer is: [LT^-2], [L], and [T]