A particle of mass m is projected upwards with velocity `v=(v_(e))/(2)`, where `v_(e)` is the escape velocity then at the maximum height the potential energy of the particle is : (R is radius of earth and M is mass of earth)

Options

A. `(-GMm)/(2R)`
B. `(-GMm)/(4R)`
C. `(-3GMm)/(4R)`
D. `(-2GMm)/(3R)`

Answer

To find the potential energy of a particle of mass \( m \) projected upwards with a velocity \( v = \frac{v_e}{2} \) at its maximum height, we can follow these steps:

### Step 1: Understand Escape Velocity
The escape velocity \( v_e \) from the surface of the Earth is given by the formula:
\[ v_e = \sqrt{\frac{2GM}{R}} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( R \) is the radius of the Earth.

### Step 2: Substitute the Given Velocity
The particle is projected with a velocity:
\[ v = \frac{v_e}{2} = \frac{1}{2} \sqrt{\frac{2GM}{R}} = \sqrt{\frac{GM}{2R}} \]

### Step 3: Calculate Initial Kinetic Energy
The initial kinetic energy \( KE_i \) of the particle when it is projected is:
\[ KE_i = \frac{1}{2} mv^2 = \frac{1}{2} m \left(\sqrt{\frac{GM}{2R}}\right)^2 = \frac{1}{2} m \cdot \frac{GM}{2R} = \frac{mgM}{4R} \]

### Step 4: Calculate Initial Potential Energy
The initial potential energy \( PE_i \) at the surface of the Earth is given by:
\[ PE_i = -\frac{GMm}{R} \]

### Step 5: Total Initial Energy
The total initial energy \( E_i \) is the sum of the initial kinetic and potential energies:
\[ E_i = KE_i + PE_i = \frac{mgM}{4R} - \frac{GMm}{R} \]
To combine these, we can express \( PE_i \) with a common denominator:
\[ E_i = \frac{mgM}{4R} - \frac{4GMm}{4R} = \frac{mgM - 4GMm}{4R} = \frac{-3GMm}{4R} \]

### Step 6: Maximum Height and Final Potential Energy
At maximum height, the kinetic energy is zero, and all the energy is potential energy \( PE_f \):
\[ PE_f = E_i = -\frac{3GMm}{4R} \]

### Conclusion
Thus, the potential energy of the particle at its maximum height is:
\[ \boxed{-\frac{3GMm}{4R}} \]