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When a projectile is fired at an angle, it follows a path known as a parabolic trajectory. The term 'parabola of safety' derives from the concept that there is a particular curved path a projectile can take that will just avoid colliding with an object—like a conical hill—in its trajectory. To guarantee the projectile clears the mound, the minimum projection speed must be calculated taking this curve into account.

Imagine the conical mound as an obstacle that our projectile needs to fly over. The apex of the parabola—the highest point of the projectile's route—must sit above the summit of the mound. This ensures a 'safety' parabola where the projectile avoids the obstacle. In more mathematical terms, we consider the mound's shape and calculate the tangent angle at its peak, then use that angle to determine the minimum initial velocity required to clear the hill without contact.

The use of trigonometry is crucial when analyzing projectile motion around obstructions like hills or mounds. The angle of projection, often represented by \(\alpha\), is particularly important as it influences the distance and height a projectile can travel.

The angular slope of a hill, which is a key factor in determining the parabolic trajectory needed to clear it, can be found using basic trigonometric principles. In the exercise, \(\tan\alpha\) is equal to the height (\(h\)) divided by the radius (\(a\)) of the hill. Knowing this, we can calculate the angle \(\alpha\) using the inverse tangent function: \(\alpha = \tan^{-1}(\frac{h}{a})\). This angle helps us define the minimum angle of projection required for a successful clearance of the hill.

Determining the minimum speed necessary for a projectile to clear an obstacle is pivotal for not only theoretical calculations but also in practical applications such as artillery operations. The formula for the smallest speed \(v\) that will ensure the projectile clears the mound is derived from the equation \(v = \sqrt{g*h} / \sin{(\alpha)}\), where \(g\) stands for the acceleration due to gravity. It cleverly intertwines the basic principles of kinematics and trigonometry.

In the context of the exercise, once the angle of the hill \(\alpha\) is identified, we can plug the known values into the equation to find the safe projectile speed. The muzzle speed of the projectile is then compared to this minimum speed. If the actual speed of the projectile is greater, the projectile will overcome the obstacle; if it is lesser, adjustments to the projectile's speed or angle would be needed for successful clearance.