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. `1:1`
B. `4:9`
C. `2:3`
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 of 2:3, we can follow these steps:

### Step 1: Understand the relationship between initial velocity and maximum height
When a body is thrown upwards, it will rise to a maximum height where its final velocity becomes zero. The relationship between the initial velocity (u), final velocity (v), acceleration due to gravity (g), and maximum height (h) can be expressed using the equation:
\[ v^2 = u^2 - 2gh \]

### Step 2: Rearranging the equation for maximum height
At the maximum height, the final velocity \( v = 0 \). Thus, we can rearrange the equation to find the maximum height:
\[ 0 = u^2 - 2gh \]
This simplifies to:
\[ h = \frac{u^2}{2g} \]
This shows that the maximum height \( h \) is directly proportional to the square of the initial velocity \( u \).

### Step 3: Define the initial velocities
Let the initial velocities of the two bodies be \( u_1 \) and \( u_2 \). According to the problem, the ratio of their initial velocities is given as:
\[ \frac{u_1}{u_2} = \frac{2}{3} \]

### Step 4: Express the maximum heights in terms of initial velocities
Using the relationship derived in Step 2, the maximum heights \( h_1 \) and \( h_2 \) for the two bodies can be expressed as:
\[ h_1 = \frac{u_1^2}{2g} \]
\[ h_2 = \frac{u_2^2}{2g} \]

### Step 5: Find the ratio of the maximum heights
To find the ratio of the maximum heights \( \frac{h_1}{h_2} \):
\[ \frac{h_1}{h_2} = \frac{\frac{u_1^2}{2g}}{\frac{u_2^2}{2g}} = \frac{u_1^2}{u_2^2} \]

### Step 6: Substitute the ratio of initial velocities
Now substituting the ratio of the initial velocities:
\[ \frac{h_1}{h_2} = \left(\frac{u_1}{u_2}\right)^2 = \left(\frac{2}{3}\right)^2 = \frac{4}{9} \]

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