Theory of relativity reveals that mass can be converted into energy. The energy E so obtained is proportional to certain powers of mass m and the speed c of light. Guess a relation among the quantities using the method of dimensions.

Explanation

To derive a relation among energy (E), mass (m), and the speed of light (c) using dimensional analysis, we can follow these steps:

### Step 1: Identify the dimensions of the quantities
- **Mass (m)** has the dimension: \([M^1 L^0 T^0]\)
- **Speed (c)** has the dimension: \([M^0 L^1 T^{-1}]\)
- **Energy (E)** can be derived from kinetic energy, which is given by the formula \(E = \frac{1}{2} mv^2\).

### Step 2: Find the dimension of energy (E)
Using the kinetic energy formula:
- The dimension of kinetic energy \(E\) is:
\[ E = \frac{1}{2} mv^2 \]
where \(v\) is velocity. The dimension of \(v\) is \([L^1 T^{-1}]\). Therefore:
\[ E = m \cdot v^2 \implies E = m \cdot (v^2) = m \cdot (L^1 T^{-1})^2 = m \cdot (L^2 T^{-2}) \]
Thus, the dimension of energy \(E\) is:
\[ [E] = [M^1 L^0 T^0] \cdot [L^2 T^{-2}] = [M^1 L^2 T^{-2}] \]

### Step 3: Set up the relation using dimensional analysis
Assume a relation of the form:
\[ E = k \cdot m^a \cdot c^b \]
where \(k\) is a dimensionless constant, and \(a\) and \(b\) are the powers we need to determine.

### Step 4: Write the dimensions of the right-hand side
The dimensions of \(m^a\) and \(c^b\) are:
- \(m^a\) has dimensions \([M^a L^0 T^0]\)
- \(c^b\) has dimensions \([M^0 L^b T^{-b}]\)

Thus, the combined dimensions of the right-hand side are:
\[ [M^a L^0 T^0] \cdot [M^0 L^b T^{-b}] = [M^a L^b T^{-b}] \]

### Step 5: Equate the dimensions
Now we equate the dimensions of both sides:
\[ [M^1 L^2 T^{-2}] = [M^a L^b T^{-b}] \]

From this, we can compare the powers of \(M\), \(L\), and \(T\):
1. For mass \(M\): \(a = 1\)
2. For length \(L\): \(b = 2\)
3. For time \(T\): \(-b = -2 \implies b = 2\)

### Step 6: Write the final relation
Substituting the values of \(a\) and \(b\) back into the equation gives us:
\[ E = k \cdot m^1 \cdot c^2 \implies E = k \cdot m \cdot c^2 \]

### Conclusion
The relation among energy, mass, and the speed of light is:
\[ E = k \cdot m c^2 \]
where \(k\) is a dimensionless constant.

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