What will be the escape speed from earth if the mass of earth is doubled keeping the same size ?

Explanation

To find the escape speed from Earth when its mass is doubled while keeping its size the same, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for Escape Speed**:
The escape speed \( v \) from a planet is given by the formula:
\[ v = \sqrt{\frac{2GM}{R}} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the planet, and \( R \) is the radius of the planet.

2. **Identify Initial Conditions**:
Let the initial mass of the Earth be \( M \) and the radius be \( R \). The initial escape speed \( v_1 \) can be expressed as:
\[ v_1 = \sqrt{\frac{2GM}{R}} \]

3. **Change the Mass of the Earth**:
If the mass of the Earth is doubled, the new mass \( M' \) becomes:
\[ M' = 2M \]
The radius remains the same, \( R \).

4. **Calculate the New Escape Speed**:
The new escape speed \( v_2 \) with the doubled mass is:
\[ v_2 = \sqrt{\frac{2GM'}{R}} = \sqrt{\frac{2G(2M)}{R}} = \sqrt{\frac{4GM}{R}} = 2\sqrt{\frac{GM}{R}} = 2v_1 \]

5. **Conclusion**:
Therefore, if the mass of the Earth is doubled while keeping its radius unchanged, the new escape speed \( v_2 \) will be:
\[ v_2 = 2v_1 \]

### Final Answer:
The escape speed from Earth, if the mass is doubled while keeping the same size, will be twice the original escape speed.

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