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Projectile motion can be described using the equations of motion for both the horizontal and vertical components. The horizontal motion of a projectile is constant and is given by the equation:

• \( x = u \cos \alpha \times t \)

This equation means that the horizontal distance covered, \( x \), is equal to the initial speed \( u \) multiplied by the cosine of the launch angle \( \alpha \), and the time \( t \).

The vertical motion is affected by gravity, hence, it follows a different pattern. The equation is:

• \( y = u \sin \alpha \times t - \frac{1}{2} g t^2 \)

Here, \( y \) is the vertical distance, \( u \sin \alpha \times t \) represents the initial vertical velocity component, and \( \frac{1}{2} g t^2 \) accounts for the effect of gravity, where \( g \) is the acceleration due to gravity.

Using these two equations, we calculate the projectile's path and determine other related parameters such as time of flight, maximum height, and range.

The trajectory of a projectile is shaped like a parabola, determined by its initial speed and the angle of projection. When a projectile is launched, it follows a curved path because of gravity pulling it down.

For a projectile to clear a barrier placed at a certain horizontal distance, it needs to be launched at an appropriate speed and angle. The position of the projectile at any given time can be calculated using the equations of motion.

At the point where the projectile needs to clear a height \( h \) placed at distance \( a \) from the launch point, we can set the vertical position \( y \) equal to \( h \). At this specific point, the horizontal distance traveled \( x \) must be \( a \).

• Time to reach \( a \): \( t = \frac{a}{u \cos \alpha} \)
• Vertical position: \( y = u \sin \alpha \frac{a}{u \cos \alpha} - \frac{1}{2} g (\frac{a}{u \cos \alpha})^2 \)

Solving this correctly ensures the projectile just skims the top of the barrier, providing a critical insight into correctly applying projectile calculations.

The initial speed \( u \), and the angle \( \alpha \), are two fundamental factors that define the projectile's path. Both must be carefully adjusted to meet specific requirements, like clearing an obstacle.

The angle of projection \( \alpha \) will determine how steep or shallow the projectile's path will be. A larger \( \alpha \) means a steeper path, while a smaller \( \alpha \) means a flatter path. The speed \( u \) influences how far and high the projectile can travel.

• A higher \( u \) increases both distance and height.

For a projectile to just clear a barrier, the speed and angle must satisfy the quadratic equation derived:

• \( \left(\frac{g a^{2}}{2 u^{2}}\right) \tan ^{2} \alpha-a \tan \alpha+\left(\frac{g a^{2}}{2 u^{2}}+h\right)=0 \)

This equation helps us determine the exact speed \( u \) that will allow the projectile to pass over the obstruction at the correct point in its trajectory. The solution incorporates solving for real values of \( \alpha \) and ensuring \( u^4 - 2ghu^2 - g^2a^2 \geq 0 \). This condition ensures that there's a viable set of speeds and angles for successful launch.