Possible directions for the same horizontal range

When a projectile is launched at a certain angle, it follows a parabolic trajectory and its range depends on the initial velocity and the angle of projection. However, there are some angles that produce the same horizontal range. Let's see how many directions are possible for the same horizontal range:

- Definition: The horizontal range is the distance traveled by a projectile in the horizontal direction before hitting the ground. It depends on the initial velocity (v) and the angle of projection (θ), and can be calculated as R = v² sin(2θ) / g, where g is the acceleration due to gravity (9.8 m/s²).
- Symmetry: The horizontal range is symmetric with respect to the angle of 45 degrees. This means that if you launch a projectile at an angle of 30 degrees and it travels a certain distance before hitting the ground, you can achieve the same distance by launching it at an angle of 60 degrees. This is because the sine function is symmetric with respect to 45 degrees.
- Complementary angles: The horizontal range is the same for two complementary angles, which add up to 90 degrees. For example, if you launch a projectile at an angle of 30 degrees and it travels a certain distance before hitting the ground, you can achieve the same distance by launching it at an angle of 60 degrees, which is the complementary angle of 30 degrees.
- Total of 2 directions: Therefore, there are only two possible directions for the same horizontal range, which are symmetric with respect to 45 degrees and add up to 90 degrees. For example, if the horizontal range is 100 meters, you can achieve it by launching the projectile at angles of 30 degrees and 60 degrees, or at angles of 40 degrees and 50 degrees, or at angles of 20 degrees and 70 degrees, and so on.

Therefore, the correct answer is option 'C', which says that there are 2 possible directions for the same horizontal range.