A particle of mass m is projected upword with velocity `v=(v_(e))/(2)` (`v_(e))` escape of the particle is

Options

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

Answer

To solve the problem of finding the potential energy of a particle projected upward with a velocity of \( v = \frac{v_e}{2} \), where \( v_e \) is the escape velocity, we can follow these steps:

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

### Step 2: Substitute for Given Velocity
In the problem, 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 \( KE \) of the particle can be calculated using the formula:
\[ KE = \frac{1}{2} mv^2 \]
Substituting for \( v \):
\[ KE = \frac{1}{2} m \left(\sqrt{\frac{GM}{2R}}\right)^2 = \frac{1}{2} m \cdot \frac{GM}{2R} = \frac{mGM}{4R} \]

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

### Step 5: Total Initial Energy
The total initial energy \( E_i \) is the sum of kinetic and potential energy:
\[ E_i = KE + PE = \frac{mGM}{4R} - \frac{GMm}{R} \]
To combine these, we need a common denominator:
\[ E_i = \frac{mGM}{4R} - \frac{4mGM}{4R} = \frac{mGM}{4R} - \frac{4mGM}{4R} = -\frac{3mGM}{4R} \]

### Step 6: Maximum Height and Final Energy
At the maximum height, the velocity of the particle becomes zero, meaning all the initial energy has converted into potential energy. Therefore, the potential energy at maximum height \( PE_{max} \) is equal to the total initial energy:
\[ PE_{max} = E_i = -\frac{3mGM}{4R} \]

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

### Answer
The potential energy of the particle at maximum height is:
\[ -\frac{3GMm}{4R} \]
This corresponds to option C.

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