The dimensional formula for energy is

Options

A. `[MLT^(-2)]`
B. `[ML^2T^(-2)]`
C. `[M^(-1) L^2T]`
D. `[ M L^2 T]`

Answer

To find the dimensional formula for energy, we can use the formula for kinetic energy as a reference. Here's a step-by-step solution:

### Step 1: Understand the formula for kinetic energy
The formula for kinetic energy (KE) is given by:
\[ KE = \frac{1}{2} mv^2 \]
where \( m \) is mass and \( v \) is velocity.

### Step 2: Identify the dimensions of mass and velocity
- The dimension of mass \( m \) is represented as \( [M] \).
- The dimension of velocity \( v \) is defined as distance per time, which can be expressed as:
\[ v = \frac{\text{distance}}{\text{time}} = \frac{L}{T} \]
Thus, the dimension of velocity \( v \) is \( [L][T^{-1}] \).

### Step 3: Calculate the dimensions of \( v^2 \)
To find the dimensions of \( v^2 \), we square the dimensions of velocity:
\[ v^2 = \left(\frac{L}{T}\right)^2 = \frac{L^2}{T^2} \]
So, the dimension of \( v^2 \) is \( [L^2][T^{-2}] \).

### Step 4: Substitute the dimensions into the kinetic energy formula
Now, substituting the dimensions of mass and \( v^2 \) into the kinetic energy formula:
\[ KE = \frac{1}{2} mv^2 \implies [KE] = [M] \cdot [L^2][T^{-2}] \]
Thus, the dimensional formula for kinetic energy becomes:
\[ [KE] = [M][L^2][T^{-2}] \]

### Step 5: Write the final dimensional formula for energy
Since kinetic energy is a form of energy, the dimensional formula for energy is:
\[ [E] = [M][L^2][T^{-2}] \]

### Conclusion
The dimensional formula for energy is:
\[ [M][L^2][T^{-2}] \]