A particle is prjected vertically upwards the surface of the earth (radius `R_(e))` with a speed equal to one fourth of escape velocity what is the maximum height attained by it from the surface of the earth?

Options

A. `(16)/(15)R_(e)`
B. `(R_(e))/(15)`
C. `(4)/(15)R_(e)`
D. None of these

Explanation

To solve the problem of finding the maximum height attained by a particle projected vertically upwards from the surface of the Earth with a speed equal to one fourth of the escape velocity, we can use the principle of conservation of energy.

### Step-by-Step Solution:

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

2. **Determine Initial Velocity**:
The particle is projected with a speed equal to one fourth of the escape velocity:
\[ V = \frac{1}{4} V_e = \frac{1}{4} \sqrt{\frac{2GM}{R_e}} \]

3. **Calculate Initial Kinetic Energy**:
The initial kinetic energy \( KE_i \) of the particle when it is at the surface of the Earth is given by:
\[ KE_i = \frac{1}{2} m V^2 = \frac{1}{2} m \left(\frac{1}{4} V_e\right)^2 = \frac{1}{2} m \left(\frac{1}{16} V_e^2\right) = \frac{m}{32} V_e^2 \]

Substituting \( V_e^2 = \frac{2GM}{R_e} \):
\[ KE_i = \frac{m}{32} \cdot \frac{2GM}{R_e} = \frac{mGM}{16R_e} \]

4. **Calculate Initial Potential Energy**:
The initial potential energy \( PE_i \) at the surface of the Earth is:
\[ PE_i = -\frac{GMm}{R_e} \]

5. **Total Initial Energy**:
The total initial energy \( E_i \) is the sum of kinetic and potential energy:
\[ E_i = KE_i + PE_i = \frac{mGM}{16R_e} - \frac{GMm}{R_e} = \frac{mGM}{16R_e} - \frac{16mGM}{16R_e} = -\frac{15mGM}{16R_e} \]

6. **At Maximum Height**:
At the maximum height \( h \), the velocity of the particle will be zero, and the potential energy will be:
\[ PE_f = -\frac{GMm}{R_e + h} \]

The total energy at maximum height \( E_f \) is:
\[ E_f = PE_f = -\frac{GMm}{R_e + h} \]

7. **Conservation of Energy**:
Setting the initial energy equal to the final energy:
\[ -\frac{15mGM}{16R_e} = -\frac{GMm}{R_e + h} \]

Canceling \( -GMm \) from both sides:
\[ \frac{15}{16R_e} = \frac{1}{R_e + h} \]

8. **Cross-Multiplying**:
\[ 15(R_e + h) = 16R_e \]
\[ 15h = 16R_e - 15R_e \]
\[ 15h = R_e \]
\[ h = \frac{R_e}{15} \]

9. **Final Height from the Surface**:
The maximum height attained from the surface of the Earth is:
\[ h = \frac{R_e}{15} \]

### Conclusion:
The maximum height attained by the particle from the surface of the Earth is \( \frac{R_e}{15} \).

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