Two bodies are thrown vertically upwards with their initial velocity in the ratio `2:3.` Then the ratio of the maximum height attained by them is

Options

A. (a)`1:1`
B. (b)`4:9`
C. (c)`2:3`
D. (d)`sqrt2:sqrt3`

Answer

To solve the problem of finding the ratio of the maximum heights attained by two bodies thrown vertically upwards with their initial velocities in the ratio 2:3, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**:
We have two bodies thrown upwards with initial velocities \( u_1 \) and \( u_2 \) such that \( \frac{u_1}{u_2} = \frac{2}{3} \).

2. **Use the Formula for Maximum Height**:
The maximum height \( h \) attained by a body thrown vertically upwards is given by the formula:
\[ h = \frac{u^2}{2g} \]
where \( u \) is the initial velocity and \( g \) is the acceleration due to gravity.

3. **Express Maximum Heights**:
For the first body with initial velocity \( u_1 \):
\[ h_1 = \frac{u_1^2}{2g} \]
For the second body with initial velocity \( u_2 \):
\[ h_2 = \frac{u_2^2}{2g} \]

4. **Find the Ratio of Maximum Heights**:
We need to find the ratio \( \frac{h_1}{h_2} \):
\[ \frac{h_1}{h_2} = \frac{\frac{u_1^2}{2g}}{\frac{u_2^2}{2g}} = \frac{u_1^2}{u_2^2} \]

5. **Substitute the Ratio of Initial Velocities**:
Since \( \frac{u_1}{u_2} = \frac{2}{3} \), we can square this ratio:
\[ \frac{h_1}{h_2} = \left(\frac{u_1}{u_2}\right)^2 = \left(\frac{2}{3}\right)^2 = \frac{4}{9} \]

6. **Conclusion**:
Therefore, the ratio of the maximum heights attained by the two bodies is:
\[ \frac{h_1}{h_2} = \frac{4}{9} \]

### Final Answer:
The ratio of the maximum heights attained by the two bodies is \( 4:9 \).

---