Two projectiles are fired from the same point with the same speed at angles of projection `60^(@) and 30^(@)` respectively. Which one of the following is true?

Options

A. Their range will be the same.
B. Their maximum height will be the same.
C. Their velocity at the heighest point will be the same.
D. Their time of flight will be the same.

Explanation

To solve the problem, we need to analyze the ranges of the two projectiles fired at angles of \(60^\circ\) and \(30^\circ\) with the same initial speed \(u\).

### Step-by-Step Solution:

1. **Understanding the Range Formula**:
The range \(R\) of a projectile is given by the formula:
\[ R = \frac{u^2 \sin(2\theta)}{g} \]
where \(u\) is the initial speed, \(\theta\) is the angle of projection, and \(g\) is the acceleration due to gravity.

2. **Calculate the Range for the First Projectile**:
For the projectile fired at an angle of \(60^\circ\):
\[ R_1 = \frac{u^2 \sin(2 \times 60^\circ)}{g} \]
We know that \(2 \times 60^\circ = 120^\circ\), so:
\[ R_1 = \frac{u^2 \sin(120^\circ)}{g} \]
Using the sine value, \(\sin(120^\circ) = \sin(180^\circ - 60^\circ) = \sin(60^\circ) = \frac{\sqrt{3}}{2}\):
\[ R_1 = \frac{u^2 \cdot \frac{\sqrt{3}}{2}}{g} = \frac{\sqrt{3} u^2}{2g} \]

3. **Calculate the Range for the Second Projectile**:
For the projectile fired at an angle of \(30^\circ\):
\[ R_2 = \frac{u^2 \sin(2 \times 30^\circ)}{g} \]
We know that \(2 \times 30^\circ = 60^\circ\), so:
\[ R_2 = \frac{u^2 \sin(60^\circ)}{g} \]
Using the sine value, \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\):
\[ R_2 = \frac{u^2 \cdot \frac{\sqrt{3}}{2}}{g} = \frac{\sqrt{3} u^2}{2g} \]

4. **Comparing the Ranges**:
From the calculations, we find:
\[ R_1 = \frac{\sqrt{3} u^2}{2g} \quad \text{and} \quad R_2 = \frac{\sqrt{3} u^2}{2g} \]
Therefore, we conclude that:
\[ R_1 = R_2 \]

5. **Conclusion**:
Since both projectiles have the same range, we can state that the ranges of the projectiles fired at angles \(60^\circ\) and \(30^\circ\) are equal.

### Final Answer:
The correct statement is that the ranges of the two projectiles are the same.

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