Three particles are projected vertically upward from a point on the surface of the earth with velocities `sqrt(2gR//3)` ,`sqrt(gR)`,`sqrt(4gR//3)` surface of the earth. The maximum heights attained are respectively `h_(1),h_(2),h_(3)`.

Options

A. `h_(1):h_(2)=2:3`
B. `h_(2):h_(3)=3:4`
C. `h_(1):h_(3)=1:4`
D. `h_(2)=R`

Explanation

To solve the problem of finding the maximum heights attained by three particles projected vertically upward from the surface of the Earth with different initial velocities, we can use the principle of conservation of mechanical energy. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Problem
We have three particles projected with the following velocities:
- \( v_1 = \sqrt{\frac{2gR}{3}} \)
- \( v_2 = \sqrt{gR} \)
- \( v_3 = \sqrt{\frac{4gR}{3}} \)

We need to find the maximum heights \( h_1, h_2, \) and \( h_3 \) attained by these particles.

### Step 2: Apply Conservation of Mechanical Energy
At the point of projection, the total mechanical energy (kinetic + potential) is conserved. The potential energy at the surface of the Earth is zero (taking it as the reference point). The kinetic energy is given by:

\[ KE = \frac{1}{2} mv^2 \]

At the maximum height, the kinetic energy becomes zero, and the potential energy is given by:

\[ PE = -\frac{GMm}{R + h} \]

Using conservation of energy:

\[ \frac{1}{2} mv^2 = -\frac{GMm}{R + h} \]

### Step 3: Simplify the Equation
Since mass \( m \) cancels out, we can rearrange the equation:

\[ \frac{v^2}{2g} = \frac{GM}{g(R + h)} \]

Using \( g = \frac{GM}{R^2} \), we can express \( GM \) as \( gR^2 \):

\[ \frac{v^2}{2g} = \frac{gR^2}{g(R + h)} \]

This simplifies to:

\[ R + h = \frac{gR^2}{v^2} \]

Thus, we can find \( h \):

\[ h = \frac{gR^2}{v^2} - R \]

### Step 4: Calculate Maximum Heights
Now we can calculate \( h_1, h_2, \) and \( h_3 \) for each velocity:

1. **For \( v_1 = \sqrt{\frac{2gR}{3}} \)**:
\[ h_1 = \frac{gR^2}{\frac{2gR}{3}} - R = \frac{3R}{2} - R = \frac{R}{2} \]

2. **For \( v_2 = \sqrt{gR} \)**:
\[ h_2 = \frac{gR^2}{gR} - R = R - R = 0 \]

3. **For \( v_3 = \sqrt{\frac{4gR}{3}} \)**:
\[ h_3 = \frac{gR^2}{\frac{4gR}{3}} - R = \frac{3R}{4} - R = -\frac{R}{4} \]

### Step 5: Ratios of Maximum Heights
Now we can find the ratios of the maximum heights:

1. **Ratio \( \frac{h_1}{h_2} \)**:
\[ \frac{h_1}{h_2} = \frac{\frac{R}{2}}{0} \text{ (undefined)} \]

2. **Ratio \( \frac{h_2}{h_3} \)**:
\[ \frac{h_2}{h_3} = \frac{0}{-\frac{R}{4}} = 0 \]

3. **Ratio \( \frac{h_1}{h_3} \)**:
\[ \frac{h_1}{h_3} = \frac{\frac{R}{2}}{-\frac{R}{4}} = -2 \]

### Conclusion
The maximum heights attained by the particles are \( h_1 = \frac{R}{2} \), \( h_2 = 0 \), and \( h_3 = -\frac{R}{4} \). The ratios of the heights can be used to determine the relationships between them.