A body is projected from a point with different angles of projections 20°, 35°, 45°, 60° with the horizontal but with same initial speed. Their respective horizontal ranges are `R_(1), R_(2), R_(3)` and `R_4`. Identify the correct order in which the horizontal ranges are arranged in increasing order

Options

A. `R_(1),R_(4),R_(2), R_(3)`
B. `R_(2),R_(1), R_(4), R_(3)`
C. `R_(1), R_(2), R_(4), R_(3)`
D. `R_(4), R_(1), R_(2), R_(3)`

Answer

To solve the problem, we need to determine the horizontal ranges of a projectile launched at different angles but with the same initial speed. The formula for the horizontal range \( R \) of a projectile is given by:

\[ R = \frac{u^2 \sin 2\theta}{g} \]

where:
- \( u \) is the initial speed,
- \( \theta \) is the angle of projection,
- \( g \) is the acceleration due to gravity.

Since \( u \) and \( g \) are constant for all projections, we can focus on the term \( \sin 2\theta \) to compare the ranges.

### Step 1: Calculate \( \sin 2\theta \) for each angle

1. **For \( \theta_1 = 20^\circ \)**:
\[ R_1 \propto \sin(2 \times 20^\circ) = \sin(40^\circ) \]
Using a calculator or trigonometric table, we find:
\[ \sin(40^\circ) \approx 0.6428 \]

2. **For \( \theta_2 = 35^\circ \)**:
\[ R_2 \propto \sin(2 \times 35^\circ) = \sin(70^\circ) \]
\[ \sin(70^\circ) \approx 0.9397 \]

3. **For \( \theta_3 = 45^\circ \)**:
\[ R_3 \propto \sin(2 \times 45^\circ) = \sin(90^\circ) \]
\[ \sin(90^\circ) = 1 \]

4. **For \( \theta_4 = 60^\circ \)**:
\[ R_4 \propto \sin(2 \times 60^\circ) = \sin(120^\circ) \]
\[ \sin(120^\circ) = \sin(180^\circ - 60^\circ) = \sin(60^\circ) \approx 0.8660 \]

### Step 2: Compare the values of \( \sin 2\theta \)

Now we have the following values:
- \( R_1 \propto \sin(40^\circ) \approx 0.6428 \)
- \( R_2 \propto \sin(70^\circ) \approx 0.9397 \)
- \( R_3 \propto \sin(90^\circ) = 1 \)
- \( R_4 \propto \sin(120^\circ) \approx 0.8660 \)

### Step 3: Arrange the ranges in increasing order

Now we can arrange the ranges based on the calculated values of \( \sin 2\theta \):
1. \( R_1 \) (for \( 20^\circ \)) \( \approx 0.6428 \)
2. \( R_4 \) (for \( 60^\circ \)) \( \approx 0.8660 \)
3. \( R_2 \) (for \( 35^\circ \)) \( \approx 0.9397 \)
4. \( R_3 \) (for \( 45^\circ \)) \( = 1 \)

Thus, the correct order in increasing range is:
\[ R_1 < R_4 < R_2 < R_3 \]

### Final Answer

The correct order in which the horizontal ranges are arranged in increasing order is:
\[ R_1, R_4, R_2, R_3 \]