The objects x and y are projected from a point with same velocities , at angle `30 ^(@) and 60^(@) `, the ratio of their maximum height attained:

Options

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

Answer

To find the ratio of the maximum heights attained by two objects projected at angles of \(30^\circ\) and \(60^\circ\) with the same initial velocity, we can follow these steps:

### Step 1: Write the formula for maximum height
The maximum height \(H\) attained by a projectile is given by the formula:
\[ H = \frac{u^2 \sin^2 \theta}{2g} \]
where:
- \(u\) is the initial velocity,
- \(\theta\) is the angle of projection,
- \(g\) is the acceleration due to gravity.

### Step 2: Define the maximum heights for both objects
Let:
- \(H_x\) be the maximum height for object X (projected at \(30^\circ\)),
- \(H_y\) be the maximum height for object Y (projected at \(60^\circ\)).

Using the formula:
\[ H_x = \frac{u^2 \sin^2(30^\circ)}{2g} \]
\[ H_y = \frac{u^2 \sin^2(60^\circ)}{2g} \]

### Step 3: Find the ratio of the maximum heights
The ratio of the maximum heights \(H_x\) to \(H_y\) is:
\[ \frac{H_x}{H_y} = \frac{\frac{u^2 \sin^2(30^\circ)}{2g}}{\frac{u^2 \sin^2(60^\circ)}{2g}} \]

### Step 4: Simplify the ratio
Since \(u^2\) and \(2g\) are common in both the numerator and the denominator, they cancel out:
\[ \frac{H_x}{H_y} = \frac{\sin^2(30^\circ)}{\sin^2(60^\circ)} \]

### Step 5: Calculate \(\sin^2(30^\circ)\) and \(\sin^2(60^\circ)\)
We know:
\[ \sin(30^\circ) = \frac{1}{2} \quad \Rightarrow \quad \sin^2(30^\circ) = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]
\[ \sin(60^\circ) = \frac{\sqrt{3}}{2} \quad \Rightarrow \quad \sin^2(60^\circ) = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} \]

### Step 6: Substitute the values into the ratio
Now substituting these values into the ratio:
\[ \frac{H_x}{H_y} = \frac{\frac{1}{4}}{\frac{3}{4}} = \frac{1}{3} \]

### Step 7: Conclusion
Thus, the ratio of the maximum heights attained by objects X and Y is:
\[ H_x : H_y = 1 : 3 \]

### Final Answer
The ratio of the maximum height attained by objects X and Y is \(1 : 3\).

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