Two projectiles thrown from the same point at angles `60^@ and 30^@` with the horizontal attain the same height. The ratio of their initial velocities is

Options

A. 1
B. 2
C. `sqrt(3)`
D. `1/sqrt3`

Answer

To solve the problem of finding the ratio of the initial velocities of two projectiles thrown at angles of \(60^\circ\) and \(30^\circ\) that attain the same height, we can follow these steps:

### Step 1: Understand the height formula in projectile motion
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, and \(g\) is the acceleration due to gravity.

### Step 2: Write the height equations for both projectiles
Let \(U_1\) be the initial velocity of the projectile thrown at \(30^\circ\) and \(U_2\) be the initial velocity of the projectile thrown at \(60^\circ\).

For the projectile at \(30^\circ\):
\[ H_1 = \frac{U_1^2 \sin^2(30^\circ)}{2g} \]
For the projectile at \(60^\circ\):
\[ H_2 = \frac{U_2^2 \sin^2(60^\circ)}{2g} \]

### Step 3: Set the heights equal to each other
Since both projectiles attain the same height:
\[ H_1 = H_2 \]
This gives us:
\[ \frac{U_1^2 \sin^2(30^\circ)}{2g} = \frac{U_2^2 \sin^2(60^\circ)}{2g} \]
We can cancel \(2g\) from both sides:
\[ U_1^2 \sin^2(30^\circ) = U_2^2 \sin^2(60^\circ) \]

### Step 4: Substitute the values of \(\sin\)
We know:
\[ \sin(30^\circ) = \frac{1}{2} \quad \text{and} \quad \sin(60^\circ) = \frac{\sqrt{3}}{2} \]
Thus,
\[ \sin^2(30^\circ) = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \quad \text{and} \quad \sin^2(60^\circ) = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} \]

### Step 5: Substitute the sine values into the equation
Now substituting these values into our equation:
\[ U_1^2 \cdot \frac{1}{4} = U_2^2 \cdot \frac{3}{4} \]
We can simplify this to:
\[ U_1^2 = U_2^2 \cdot 3 \]

### Step 6: Find the ratio of initial velocities
Taking the square root of both sides:
\[ \frac{U_1}{U_2} = \sqrt{3} \]
Thus, the ratio of the initial velocities \(U_1\) to \(U_2\) is:
\[ \frac{U_1}{U_2} = \sqrt{3} \]

### Step 7: Find the inverse ratio
Since we need the ratio of \(U_2\) to \(U_1\):
\[ \frac{U_2}{U_1} = \frac{1}{\sqrt{3}} \]

### Final Answer
The ratio of their initial velocities \(U_2:U_1\) is:
\[ \frac{U_2}{U_1} = \frac{1}{\sqrt{3}} \]