If `M_(e )` and `R_(e )` be the mass and radius of earth, then the escape velocity will be _________.

Answer

To find the escape velocity from the Earth, we can use the principle of conservation of mechanical energy. Here’s a step-by-step solution:

### Step 1: Understand Escape Velocity
Escape velocity is the minimum velocity required for an object to break free from the gravitational attraction of a celestial body, in this case, the Earth.

### Step 2: Apply Conservation of Energy
We can apply the conservation of mechanical energy between two points:
- Point A (on the surface of the Earth)
- Point B (at an infinite distance from the Earth)

The equation for conservation of energy states:
\[ U_A + K_A = U_B + K_B \]

Where:
- \( U \) is the potential energy
- \( K \) is the kinetic energy

### Step 3: Define Potential and Kinetic Energy
At point A (on the surface of the Earth):
- Potential Energy \( U_A = -\frac{G M_e m}{R_e} \)
- Kinetic Energy \( K_A = \frac{1}{2} m v_e^2 \)

At point B (at infinity):
- Potential Energy \( U_B = 0 \)
- Kinetic Energy \( K_B = 0 \)

### Step 4: Set Up the Equation
Substituting the energies into the conservation equation:
\[ -\frac{G M_e m}{R_e} + \frac{1}{2} m v_e^2 = 0 \]

### Step 5: Simplify the Equation
Rearranging the equation gives:
\[ \frac{1}{2} m v_e^2 = \frac{G M_e m}{R_e} \]

### Step 6: Cancel Out Mass
Since mass \( m \) appears on both sides, we can cancel it out:
\[ \frac{1}{2} v_e^2 = \frac{G M_e}{R_e} \]

### Step 7: Solve for Escape Velocity
Multiplying both sides by 2 gives:
\[ v_e^2 = \frac{2 G M_e}{R_e} \]

Taking the square root of both sides, we find:
\[ v_e = \sqrt{\frac{2 G M_e}{R_e}} \]

### Final Answer
Thus, the escape velocity from the Earth is:
\[ v_e = \sqrt{\frac{2 G M_e}{R_e}} \]

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