A body is projected vertically upwards with speed `sqrt(g R_(e ))` from the surface of earth. What is the maximum height attained by the body ? [`R_(e )` is the radius of earth, g is acceleration due to gravity]

Options

A. `R_(e )`
B. `2R_(e )`
C. `(R_(e ))/(2)`
D. None of these

Explanation

To solve the problem of finding the maximum height attained by a body projected vertically upwards with speed \( v = \sqrt{g R_e} \) from the surface of the Earth, we can use the principle of conservation of mechanical energy. Here’s a step-by-step solution:

### Step 1: Understand the initial conditions
The body is projected from the surface of the Earth with an initial speed \( v = \sqrt{g R_e} \). At this point, the initial kinetic energy (KE) and potential energy (PE) need to be calculated.

### Step 2: Calculate initial kinetic energy
The initial kinetic energy (KE_initial) of the body can be calculated using the formula:
\[ \text{KE}_{\text{initial}} = \frac{1}{2} mv^2 \]
Substituting \( v = \sqrt{g R_e} \):
\[ \text{KE}_{\text{initial}} = \frac{1}{2} m (\sqrt{g R_e})^2 = \frac{1}{2} m g R_e \]

### Step 3: Calculate initial potential energy
The initial potential energy (PE_initial) when the body is at the surface of the Earth is given by:
\[ \text{PE}_{\text{initial}} = -\frac{G M m}{R_e} \]
where \( G \) is the gravitational constant and \( M \) is the mass of the Earth. We can express \( g \) as \( g = \frac{G M}{R_e^2} \), thus:
\[ \text{PE}_{\text{initial}} = -\frac{G M m}{R_e} = -\frac{g m R_e}{g} = -m g R_e \]

### Step 4: Set up the conservation of energy equation
At the maximum height \( h \), the body will momentarily come to rest, so its kinetic energy will be zero. The potential energy at height \( h \) is given by:
\[ \text{PE}_{\text{final}} = -\frac{G M m}{R_e + h} \]
Using conservation of mechanical energy:
\[ \text{KE}_{\text{initial}} + \text{PE}_{\text{initial}} = \text{KE}_{\text{final}} + \text{PE}_{\text{final}} \]
Substituting the values:
\[ \frac{1}{2} m g R_e - m g R_e = 0 - \frac{G M m}{R_e + h} \]

### Step 5: Simplify the equation
This simplifies to:
\[ \frac{1}{2} m g R_e - m g R_e = -\frac{G M m}{R_e + h} \]
\[ -\frac{1}{2} m g R_e = -\frac{G M m}{R_e + h} \]
Cancelling \( m \) from both sides (assuming \( m \neq 0 \)):
\[ \frac{1}{2} g R_e = \frac{G M}{R_e + h} \]

### Step 6: Substitute \( g \) and solve for \( h \)
Substituting \( g = \frac{G M}{R_e^2} \):
\[ \frac{1}{2} \frac{G M}{R_e^2} R_e = \frac{G M}{R_e + h} \]
This simplifies to:
\[ \frac{1}{2} \frac{G M}{R_e} = \frac{G M}{R_e + h} \]
Cancelling \( G M \) from both sides (assuming \( G M \neq 0 \)):
\[ \frac{1}{2} \frac{1}{R_e} = \frac{1}{R_e + h} \]

### Step 7: Cross-multiply and solve for \( h \)
Cross-multiplying gives:
\[ R_e + h = 2 R_e \]
Thus,
\[ h = 2 R_e - R_e = R_e \]

### Final Answer
The maximum height attained by the body is:
\[ \boxed{R_e} \]