The dimensional formul for impulse is same as the dimensional formula for

Options

A. momentum
B. force
C. rate of change of momentum
D. torque

Explanation

To solve the question regarding the dimensional formula for impulse and its comparison with other physical quantities, we can follow these steps:

### Step 1: Understand the Definition of Impulse
Impulse is defined as the product of force and the time duration over which the force acts. Mathematically, it can be expressed as:
\[ \text{Impulse} = \text{Force} \times \text{Time} \]

### Step 2: Find the Dimensional Formula for Force
The dimensional formula for force (F) is derived from Newton's second law, which states that force is equal to mass times acceleration. The dimensional formula for mass (M) is \( [M] \), and for acceleration (a), which is the change in velocity (length/time) per unit time, it is:
\[ [a] = \frac{[L]}{[T^2]} \]
Thus, the dimensional formula for force is:
\[ [F] = [M][a] = [M][L][T^{-2}] = [M L T^{-2}] \]

### Step 3: Find the Dimensional Formula for Time
The dimensional formula for time (t) is simply:
\[ [T] \]

### Step 4: Combine the Dimensional Formulas to Find Impulse
Now, substituting the dimensional formulas for force and time into the impulse formula:
\[ [\text{Impulse}] = [F] \times [T] = [M L T^{-2}] \times [T] = [M L T^{-1}] \]

### Step 5: Compare with Other Quantities
Now, we need to compare the dimensional formula of impulse \( [M L T^{-1}] \) with the given options:

1. **Momentum**:
- Momentum is defined as mass times velocity:
\[ \text{Momentum} = \text{Mass} \times \text{Velocity} \]
- The dimensional formula for velocity is \( [L T^{-1}] \), so:
\[ [\text{Momentum}] = [M] \times [L T^{-1}] = [M L T^{-1}] \]
- This matches the dimensional formula for impulse.

2. **Force**:
- As calculated earlier, the dimensional formula for force is:
\[ [F] = [M L T^{-2}] \]
- This does not match the dimensional formula for impulse.

3. **Rate of Change of Momentum**:
- The rate of change of momentum is momentum per unit time:
\[ \text{Rate of Change of Momentum} = \frac{\text{Momentum}}{t} \]
- Therefore:
\[ [\text{Rate of Change of Momentum}] = \frac{[M L T^{-1}]}{[T]} = [M L T^{-2}] \]
- This does not match the dimensional formula for impulse.

4. **Torque**:
- Torque is defined as force times distance:
\[ \text{Torque} = \text{Force} \times \text{Distance} \]
- Thus:
\[ [\text{Torque}] = [F] \times [L] = [M L T^{-2}] \times [L] = [M L^2 T^{-2}] \]
- This does not match the dimensional formula for impulse.

### Conclusion
From the comparisons, we find that the only quantity with the same dimensional formula as impulse is momentum.

### Final Answer
The dimensional formula for impulse is the same as that of **momentum**.