What is the maximum height attained by a body projected with a velocity equal to one- third of the escape velocity from the surface of the earth? (Radius of the earth=R)

Options

A. `R//2`
B. `R//3`
C. `R//5`
D. `R//8`

Answer

To find the maximum height attained by a body projected with a velocity equal to one-third of the escape velocity from the surface of the Earth, we can follow these steps:

### Step 1: Calculate 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 universal gravitational constant, \( M \) is the mass of the Earth, and \( R \) is the radius of the Earth.

### Step 2: Determine the Given Velocity
The body is projected with a velocity \( v \) equal to one-third of the escape velocity:
\[ v = \frac{1}{3} v_e = \frac{1}{3} \sqrt{\frac{2GM}{R}} \]

### Step 3: Calculate Initial Kinetic Energy
The initial kinetic energy \( KE \) of the body can be calculated using the formula:
\[ KE = \frac{1}{2} mv^2 \]
Substituting \( v \):
\[ KE = \frac{1}{2} m \left(\frac{1}{3} \sqrt{\frac{2GM}{R}}\right)^2 = \frac{1}{2} m \left(\frac{1}{9} \cdot \frac{2GM}{R}\right) = \frac{mGM}{9R} \]

### Step 4: Calculate Gravitational Potential Energy at Maximum Height
At the maximum height \( h \), the gravitational potential energy \( PE \) is given by:
\[ PE = -\frac{GMm}{R+h} \]
At the maximum height, all the kinetic energy will be converted to gravitational potential energy. Thus, we set the initial kinetic energy equal to the change in potential energy:
\[ \frac{mGM}{9R} = \left(-\frac{GMm}{R+h}\right) - \left(-\frac{GMm}{R}\right) \]
This simplifies to:
\[ \frac{mGM}{9R} = \frac{GMm}{R} - \frac{GMm}{R+h} \]

### Step 5: Simplify the Equation
Cancelling \( GMm \) from both sides gives:
\[ \frac{1}{9R} = \frac{1}{R} - \frac{1}{R+h} \]
Rearranging this leads to:
\[ \frac{1}{R+h} = \frac{1}{R} - \frac{1}{9R} = \frac{9-1}{9R} = \frac{8}{9R} \]
Taking the reciprocal gives:
\[ R+h = \frac{9R}{8} \]

### Step 6: Solve for Maximum Height
Now, we can find \( h \):
\[ h = \frac{9R}{8} - R = \frac{9R - 8R}{8} = \frac{R}{8} \]

Thus, the maximum height attained by the body is:
\[ \boxed{\frac{R}{8}} \]