Rank the launch angles for the five path in the figure below with respect to time of flight, from the shortest time of fight the longest :-

Options

A. `15^(@),30^(@),45^(@),60^(@),75^(@)`
B. `75^(@),60^(@),45^(@),30^(@),15^(@)`
C. `15^(@),75^(@),30^(@),60^(@),45^(@)`
D. `30^(@),60^(@),15^(@),45^(@),75^(@)`

Answer

To solve the problem of ranking the launch angles with respect to time of flight from shortest to longest, we can follow these steps:

### Step 1: Understand the formula for time of flight
The time of flight \( T \) for a projectile launched at an angle \( \theta \) is given by the formula:
\[ T = \frac{2a \sin \theta}{g} \]
where:
- \( a \) is the initial velocity,
- \( g \) is the acceleration due to gravity,
- \( \theta \) is the launch angle.

### Step 2: Analyze the sine function
The sine function \( \sin \theta \) varies with the angle \( \theta \):
- \( \sin 0^\circ = 0 \)
- \( \sin 15^\circ \)
- \( \sin 30^\circ = \frac{1}{2} \)
- \( \sin 45^\circ = \frac{\sqrt{2}}{2} \)
- \( \sin 60^\circ = \frac{\sqrt{3}}{2} \)
- \( \sin 75^\circ \)

As the angle increases from \( 0^\circ \) to \( 90^\circ \), the value of \( \sin \theta \) increases. Therefore, the time of flight \( T \) will also increase with increasing \( \theta \).

### Step 3: List the angles and their sine values
- \( 15^\circ \) → \( \sin 15^\circ \)
- \( 30^\circ \) → \( \sin 30^\circ = 0.5 \)
- \( 45^\circ \) → \( \sin 45^\circ = \frac{\sqrt{2}}{2} \approx 0.707 \)
- \( 60^\circ \) → \( \sin 60^\circ = \frac{\sqrt{3}}{2} \approx 0.866 \)
- \( 75^\circ \) → \( \sin 75^\circ \)

### Step 4: Rank the angles based on sine values
Since \( T \) is directly proportional to \( \sin \theta \), we can rank the angles based on the sine values:
- The smallest sine value corresponds to the smallest angle, which will have the shortest time of flight.
- The largest sine value corresponds to the largest angle, which will have the longest time of flight.

### Step 5: Determine the order
From the sine values, we can determine the order:
1. \( 15^\circ \) (smallest)
2. \( 30^\circ \)
3. \( 45^\circ \)
4. \( 60^\circ \)
5. \( 75^\circ \) (largest)

### Final Ranking
Thus, the ranking of the launch angles from the shortest time of flight to the longest is:
1. \( 15^\circ \)
2. \( 30^\circ \)
3. \( 45^\circ \)
4. \( 60^\circ \)
5. \( 75^\circ \)

### Conclusion
The correct answer is that the launch angles ranked from shortest to longest time of flight are:
- \( 15^\circ, 30^\circ, 45^\circ, 60^\circ, 75^\circ \)