With what velocity a body of mass m be projected vertically upwards so that it may be able to reach a height nR above earth's surface

Explanation

To find the velocity with which a body of mass \( m \) must be projected vertically upwards to reach a height \( nR \) above the Earth's surface, we can use the principle of conservation of energy. Here’s a step-by-step solution:

### Step 1: Understand the Conservation of Energy
The conservation of energy states that the total mechanical energy (kinetic energy + potential energy) of the body remains constant if only conservative forces are acting on it.

### Step 2: Write the Energy Equation
The initial total energy (at the Earth's surface) can be expressed as:
\[ E_i = U_i + K_i \]
Where:
- \( U_i \) = Initial Potential Energy = \(-\frac{GMm}{R}\)
- \( K_i \) = Initial Kinetic Energy = \(\frac{1}{2}mv^2\)

The final total energy (at height \( nR \)) can be expressed as:
\[ E_f = U_f + K_f \]
Where:
- \( U_f \) = Final Potential Energy = \(-\frac{GMm}{R + nR} = -\frac{GMm}{(1+n)R}\)
- \( K_f \) = Final Kinetic Energy = \(0\) (since the body comes to rest at the maximum height)

### Step 3: Set Up the Energy Conservation Equation
According to the conservation of energy:
\[ E_i = E_f \]
Substituting the expressions for \( E_i \) and \( E_f \):
\[ -\frac{GMm}{R} + \frac{1}{2}mv^2 = -\frac{GMm}{(1+n)R} \]

### Step 4: Rearranging the Equation
Rearranging gives:
\[ \frac{1}{2}mv^2 = -\frac{GMm}{(1+n)R} + \frac{GMm}{R} \]
Factoring out \( GMm \):
\[ \frac{1}{2}mv^2 = GMm\left(\frac{1}{R} - \frac{1}{(1+n)R}\right) \]
\[ \frac{1}{2}mv^2 = GMm\left(\frac{(1+n) - 1}{(1+n)R}\right) \]
\[ \frac{1}{2}mv^2 = \frac{GMmn}{(1+n)R} \]

### Step 5: Solve for \( v^2 \)
Dividing both sides by \( m \) and multiplying by 2:
\[ v^2 = \frac{2GMn}{(1+n)R} \]

### Step 6: Substitute \( g \)
Using the relation \( g = \frac{GM}{R^2} \), we can rewrite \( GM \) as \( gR^2 \):
\[ v^2 = \frac{2gRn}{(1+n)} \]

### Step 7: Take the Square Root
Taking the square root gives:
\[ v = \sqrt{\frac{2gRn}{(1+n)}} \]

### Final Answer
Thus, the velocity \( v \) with which the body must be projected is:
\[ v = \sqrt{\frac{2gRn}{(1+n)}} \]