Two projectiles are projected with the same velocity. If one is projected at an angle of `30^(@)` and the other at `60^(@)` to the horizontal, then ratio of maximum heights reached, is

Options

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

Answer

To solve the problem of finding the ratio of maximum heights reached by two projectiles projected at angles of \(30^\circ\) and \(60^\circ\) with the same initial velocity, we can follow these steps:

### Step 1: Understand the formula for maximum height
The maximum height \(H\) reached 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 heights for both projectiles
Let:
- \(H_1\) be the maximum height for the projectile projected at \(30^\circ\),
- \(H_2\) be the maximum height for the projectile projected at \(60^\circ\).

Using the formula, we can express the heights as:
\[ H_1 = \frac{u^2 \sin^2(30^\circ)}{2g} \]
\[ H_2 = \frac{u^2 \sin^2(60^\circ)}{2g} \]

### Step 3: Calculate \(\sin^2(30^\circ)\) and \(\sin^2(60^\circ)\)
We know:
- \(\sin(30^\circ) = \frac{1}{2}\)
- \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\)

Now, squaring these values:
\[ \sin^2(30^\circ) = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]
\[ \sin^2(60^\circ) = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} \]

### Step 4: Substitute the values into the height formulas
Now substituting these values back into the height formulas:
\[ H_1 = \frac{u^2 \cdot \frac{1}{4}}{2g} = \frac{u^2}{8g} \]
\[ H_2 = \frac{u^2 \cdot \frac{3}{4}}{2g} = \frac{3u^2}{8g} \]

### Step 5: Find the ratio of maximum heights
Now, we can find the ratio of the maximum heights:
\[ \frac{H_1}{H_2} = \frac{\frac{u^2}{8g}}{\frac{3u^2}{8g}} = \frac{1}{3} \]

### Conclusion
The ratio of the maximum heights reached by the two projectiles is:
\[ \frac{H_1}{H_2} = \frac{1}{3} \]

### Final Answer
Thus, the ratio of maximum heights reached by the projectiles projected at \(30^\circ\) and \(60^\circ\) is \(1:3\).

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