Final answer:
The angle of projection is determined by utilizing the properties of projectile motion, focusing on the relationship between the horizontal range and the maximum height, with consideration to the fact that the range is maximum at 45° and that complementary angles produce the same range.
Step-by-step explanation:
To find the angle of projection when the horizontal range is 9.5 times the vertical height, we can use the properties of projectile motion. We know that the range R is maximum at 45° provided air resistance is neglected. Considering that the range R of a projectile on level ground is determined by the launch angle θ and the initial speed v0, the formula for R is R = (v0² · sin(2θ)) / g, where g is the acceleration due to gravity.
Since we are given the ratio of the range to the maximum height, we can relate them using the fact that angles that sum to 90° will produce the same range. For a projectile with an initial speed v0, the time taken to reach the maximum height H is t = (v0 · sin(θ)) / g and H can be given by H = (v0² · sin2(θ)) / (2g). From the range equation and the height equation, we can derive a relationship between the angle and the given ratio to find the two possible angles. The angle of projection that satisfies the given condition is therefore calculated using trigonometric identities and the kinematic equations for projectile motion.
The range is 9.5 times the maximum height informs that we are dealing with complementary angles, suggesting that one of the angles is likely below 45°, considering that the range for an angle above 45° would result in a larger height than the range.