Einstein in `1905` proppunded the special theory of relativity and in `1915` proposed the general theory of relativity. The special theory deals with inertial frames of reference. The general theory of relativity deals with problems in which one frame of reference. He assumed that fixed frame is accelerated w.r.t. another frame of reference of reference cannot be located. Postulated of special theory of realtivity
● The laws of physics have the same form in all inertial systems.
● The velocity light in empty space is a unicersal constant the same for all observers.
● Einstein proved the following facts based on his theory of special relativity. Let v be the velocity of the speceship w.r.t a given frame of reference. The obserations are made by an observer in that reference frame.
● All clocks on the spaceship wil go slow by a factor `sqrt(1-v^(2)//c^(2))`
● All objects on the spaceship will have contracted in length by a factor `sqrt(1-v^(2)//c^(2))`
● The mass of the spaceship increases by a factor `sqrt(1-v^(2)//c^(2))`
● Mass and energy are interconvertable `E = mc^(2)`
The speed of a meterial object can never exceed the velocity of light.
● If two objects A and B are moving with velocity u and v w.r.t each other along the `x`-axis, the relative velocity of A w.r.t. `B = (u-v)/(1-uv//v^(2))`
A stationary body explodes into two fragments each of rest mass `1 kg` that move apart at speed of `0.6c` relative to the original body. The rest mass of the original body is -

Options

A. `2 lg`
B. `2.5 kg`
C. `1.6 kg`
D. `2.25 kg`

Explanation

To solve the problem of finding the rest mass of the original body that explodes into two fragments, we can use the principles of conservation of energy and the effects of special relativity.

### Step-by-Step Solution:

1. **Understanding the Situation**:
- We have a stationary body that explodes into two fragments, each with a rest mass of \( m_0 = 1 \, \text{kg} \).
- The fragments move apart at a speed of \( 0.6c \) relative to the original body.

2. **Using Conservation of Energy**:
- According to the conservation of energy in the context of special relativity, the total energy before the explosion must equal the total energy after the explosion.
- The energy of the original body can be expressed as \( E = mc^2 \), where \( m \) is the rest mass of the original body.

3. **Calculating the Energy of the Fragments**:
- Each fragment has a rest mass of \( 1 \, \text{kg} \) and is moving at a speed of \( 0.6c \).
- The relativistic mass of each fragment can be calculated using the formula:
\[ m' = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}} \]
- Here, \( v = 0.6c \), so:
\[ m' = \frac{1 \, \text{kg}}{\sqrt{1 - (0.6)^2}} = \frac{1 \, \text{kg}}{\sqrt{1 - 0.36}} = \frac{1 \, \text{kg}}{\sqrt{0.64}} = \frac{1 \, \text{kg}}{0.8} = 1.25 \, \text{kg} \]

4. **Total Energy of the Fragments**:
- Since there are two fragments, the total relativistic mass is:
\[ 2m' = 2 \times 1.25 \, \text{kg} = 2.5 \, \text{kg} \]
- The total energy of the fragments can be calculated as:
\[ E_{\text{fragments}} = 2m'c^2 = 2.5 \, \text{kg} \cdot c^2 \]

5. **Setting Up the Equation**:
- According to conservation of energy:
\[ mc^2 = 2.5 \, \text{kg} \cdot c^2 \]
- The \( c^2 \) terms cancel out, leading to:
\[ m = 2.5 \, \text{kg} \]

6. **Conclusion**:
- The rest mass of the original body is \( 2.5 \, \text{kg} \).

### Final Answer:
The rest mass of the original body is **2.5 kg**.

---