The position of a particle at time `t` is given by the relation `x(t) = ( v_(0) /( alpha)) ( 1 - c^(-at))`, where `v_(0)` is a constant and `alpha gt 0`. Find the dimensions of `v_(0) and alpha`.

Options

A. `M^(0) LT^(-1) and T^(-1)`
B. `M^(0) LT^(1) and T^(-1)`
C. `M^(0) LT^(-1) and LT^(-2)`
D. `M^(0) LT^(-1) and T`

Explanation

To find the dimensions of \( v_0 \) and \( \alpha \) in the given position equation \( x(t) = \frac{v_0}{\alpha} (1 - c^{-at}) \), we will analyze the equation step by step.

### Step 1: Identify the dimensions of position
The left-hand side of the equation represents the position \( x(t) \). The dimension of position is given by:
\[ [x] = L \]
where \( L \) represents length.

### Step 2: Analyze the right-hand side of the equation
The right-hand side of the equation is \( \frac{v_0}{\alpha} (1 - c^{-at}) \). Since \( 1 \) is dimensionless, the term \( \frac{v_0}{\alpha} \) must also be dimensionless. Therefore:
\[ \left[ \frac{v_0}{\alpha} \right] = 1 \]

### Step 3: Set up the relationship for dimensions
From the above, we can express the dimensions as:
\[ [v_0] = [\alpha] \]
This means that the dimensions of \( v_0 \) must be equal to the dimensions of \( \alpha \) multiplied by a dimensionless quantity.

### Step 4: Analyze the term \( c^{-at} \)
The term \( c^{-at} \) must also be dimensionless. Since \( a \) is a constant, we can analyze the dimensions of \( at \):
\[ [at] = 1 \quad \text{(dimensionless)} \]
This implies that the dimension of \( a \) must be the inverse of the dimension of time:
\[ [a] = T^{-1} \]
Thus, we can conclude that \( \alpha \) must also have the dimension of \( T^{-1} \).

### Step 5: Determine the dimensions of \( v_0 \)
Now, since we established that:
\[ [v_0] = [\alpha] \cdot [L] \]
and knowing that \( [\alpha] = T^{-1} \), we can express the dimensions of \( v_0 \) as:
\[ [v_0] = L \cdot T^{-1} \]

### Final Results
Thus, we have:
- The dimensions of \( v_0 \) are \( [v_0] = LT^{-1} \)
- The dimensions of \( \alpha \) are \( [\alpha] = T^{-1} \)

### Summary of Dimensions
- \( v_0 \) has dimensions of speed (length per time).
- \( \alpha \) has dimensions of frequency (inverse time).