Velocity of a particle at time t is expressed as follows :
`v= alpha t+(beta)/(t +gamma)`
Dimensions of `alpha, beta " and "gamma` are respectively.

Options

A. `L^(3),T,LT^(-1)`
B. `LT^(-2),L,T`
C. `L,T,LT^(-1)`
D. None of these

Explanation

To find the dimensions of the constants \( \alpha \), \( \beta \), and \( \gamma \) in the equation for velocity given by:

\[ v = \alpha t + \frac{\beta}{t + \gamma} \]

we will analyze each term in the equation step by step.

### Step 1: Identify the dimensions of velocity
The dimension of velocity \( v \) is given by:

\[ [v] = L T^{-1} \]

where \( L \) represents length and \( T \) represents time.

### Step 2: Analyze the term \( \alpha t \)
In the term \( \alpha t \), the dimensions must match the dimensions of velocity. Therefore, we can write:

\[ [v] = [\alpha][t] \]

Substituting the known dimensions:

\[ L T^{-1} = [\alpha] \cdot [T] \]

To isolate \( [\alpha] \), we rearrange the equation:

\[ [\alpha] = \frac{L T^{-1}}{[T]} = L T^{-2} \]

Thus, the dimension of \( \alpha \) is:

\[ [\alpha] = L T^{-2} \]

### Step 3: Analyze the term \( \frac{\beta}{t + \gamma} \)
For the term \( \frac{\beta}{t + \gamma} \) to have the same dimensions as velocity, we need to analyze the dimensions of \( \beta \).

Since \( t \) has dimensions of time \( [T] \) and \( \gamma \) also has dimensions of time \( [T] \), the term \( t + \gamma \) has dimensions of time \( [T] \).

Thus, we can write:

\[ [v] = \frac{[\beta]}{[T]} \]

Substituting the dimensions of velocity:

\[ L T^{-1} = \frac{[\beta]}{[T]} \]

Rearranging gives:

\[ [\beta] = L T^{-1} \cdot [T] = L \]

So, the dimension of \( \beta \) is:

\[ [\beta] = L \]

### Step 4: Analyze the dimension of \( \gamma \)
Since \( \gamma \) is added to \( t \) in the term \( t + \gamma \), it must have the same dimensions as time. Therefore:

\[ [\gamma] = [T] \]

### Summary of dimensions
We have found the dimensions of \( \alpha \), \( \beta \), and \( \gamma \):

- \( [\alpha] = L T^{-2} \)
- \( [\beta] = L \)
- \( [\gamma] = T \)

### Final Answer
The dimensions of \( \alpha \), \( \beta \), and \( \gamma \) are respectively:

\[ [\alpha] = L T^{-2}, \quad [\beta] = L, \quad [\gamma] = T \]