Three particles are projected vertically upward from a point on the surface of earth with velocities
`v_(1)=sqrt((2gR)/(3)),v_(2)sqrt(gR),v_(3)sqrt((4gR)/(3))`
respectively, where g is acceleation due to gravity on the surface of earth. If the maximum height attained are `h_(1),h_(2) " and " h_(3)` respectively, then `h_(1):h_(2):h_(3)` is

Options

A. `1:2:3`
B. `2:3:4`
C. `1:2:4`
D. `1:3:5`

Explanation

To solve the problem of finding the ratio of maximum heights attained by three particles projected vertically upward with given velocities, we can use the principle of conservation of energy. Here’s a step-by-step solution:

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

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

### Step 2: Use the Conservation of Energy
The maximum height \( h \) reached by a particle can be found using the conservation of energy principle, where the initial kinetic energy is converted to gravitational potential energy at the maximum height.

The kinetic energy (KE) at the surface is given by:
\[ KE = \frac{1}{2} mv^2 \]
The gravitational potential energy (PE) at height \( h \) is given by:
\[ PE = mgh \]

At the maximum height, all kinetic energy is converted into potential energy:
\[ \frac{1}{2} mv^2 = mgh \]

### Step 3: Solve for Maximum Height
From the equation above, we can cancel \( m \) (mass of the particle) from both sides:
\[ \frac{1}{2} v^2 = gh \]
Rearranging gives:
\[ h = \frac{v^2}{2g} \]

### Step 4: Calculate Maximum Heights for Each Particle
1. For \( v_1 = \sqrt{\frac{2gR}{3}} \):
\[ h_1 = \frac{(\sqrt{\frac{2gR}{3}})^2}{2g} = \frac{\frac{2gR}{3}}{2g} = \frac{R}{3} \]

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

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

### Step 5: Find the Ratios
Now we have the heights:
- \( h_1 = \frac{R}{3} \)
- \( h_2 = \frac{R}{2} \)
- \( h_3 = \frac{2R}{3} \)

To find the ratio \( h_1:h_2:h_3 \), we express them with a common denominator:
- \( h_1 = \frac{R}{3} = \frac{2R}{6} \)
- \( h_2 = \frac{R}{2} = \frac{3R}{6} \)
- \( h_3 = \frac{2R}{3} = \frac{4R}{6} \)

Thus, the ratio becomes:
\[ h_1:h_2:h_3 = 2:3:4 \]

### Final Answer
The ratio of the maximum heights attained by the three particles is:
\[ \boxed{2:3:4} \]