If momentum `(p)`, area `(A)` and time`(t) `are taken to be fundamental quantities then energy has the dimensional formula

Options

A. `[P^(1//2)"AT"^(-1)] `
B. `[PA^(1//2)T^(-1)] `
C. `[PA^(1//2)T^(-1)] `
D. `[P^(2)AT^(-2)] `

Explanation

To find the dimensional formula for energy when momentum (p), area (A), and time (t) are taken as fundamental quantities, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the formula for energy**:
We can use the formula for potential energy, which is given by:
\[ E = mgh \]
where:
- \( m \) = mass
- \( g \) = acceleration due to gravity
- \( h \) = height

2. **Write the dimensions of each quantity**:
- The dimensional formula for mass \( m \) is \( [M] \).
- The dimensional formula for acceleration due to gravity \( g \) is \( [L][T^{-2}] \).
- The dimensional formula for height \( h \) is \( [L] \).

3. **Combine the dimensions for potential energy**:
Substituting the dimensions into the formula for potential energy:
\[ [E] = [M][L][T^{-2}][L] = [M][L^2][T^{-2}] \]
Thus, the dimensional formula for energy \( E \) is:
\[ [E] = [M][L^2][T^{-2}] \]

4. **Express energy in terms of the fundamental quantities**:
Since we are given that momentum \( p \), area \( A \), and time \( t \) are fundamental quantities, we need to express energy in terms of these quantities.
- The dimensional formula for momentum \( p \) is:
\[ [p] = [M][L][T^{-1}] \]
- The dimensional formula for area \( A \) is:
\[ [A] = [L^2] \]
- The dimensional formula for time \( t \) is:
\[ [t] = [T] \]

5. **Set up the relationship**:
We can express energy in terms of these fundamental quantities as:
\[ [E] = k[A^a][p^b][t^c] \]
where \( k \) is a dimensionless constant and \( a, b, c \) are the powers of area, momentum, and time respectively.

6. **Substitute the dimensions**:
Substituting the dimensions into the equation gives:
\[ [E] = k[L^2]^a [M L T^{-1}]^b [T]^c \]
This simplifies to:
\[ [E] = k[M^b][L^{2a + b}][T^{-b + c}] \]

7. **Equate the dimensions**:
Now, we equate the dimensions from our earlier calculation \( [E] = [M][L^2][T^{-2}] \):
- For mass: \( b = 1 \)
- For length: \( 2a + b = 2 \)
- For time: \( -b + c = -2 \)

8. **Solve the equations**:
From \( b = 1 \):
- Substitute \( b \) into \( 2a + 1 = 2 \):
\[ 2a = 1 \implies a = \frac{1}{2} \]
- Substitute \( b \) into \( -1 + c = -2 \):
\[ c = -1 \]

9. **Final expression for energy**:
Thus, we can express the dimensional formula for energy as:
\[ [E] = k[A^{1/2}][p^1][t^{-1}] \]

### Final Answer:
The dimensional formula for energy in terms of momentum, area, and time is:
\[ [E] = k \cdot A^{1/2} \cdot p^1 \cdot t^{-1} \]