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In projectile motion, gravity plays a crucial role. It affects how quickly objects drop vertically when they are otherwise moving horizontally. For any object in freefall or moving along a curved trajectory, it experiences an acceleration due to gravity, represented as \( g \). This gravitational effect causes the bullet in the exercise to drop below the bull's-eye target, even though it is aimed directly at it from a horizontal perspective. Gravity accelerates all objects downwards at a constant rate, approximately \(9.81 \text{m/s}^2\) on Earth. Therefore, while the bullet travels horizontally towards the target, gravity pulls it downward incrementally. This is why in the equation \( y = \frac{1}{2} g t^2 \), \( y \) represents the vertical displacement, demonstrating how gravity increasingly influences the projectile over time. To predict or measure this vertical displacement, understanding and applying the concept of gravity's constant acceleration on an object is essential.

Projectile motion involves two distinct components: horizontal and vertical motion. Understanding how they work independently yet together helps solve problems like the bullet hitting below the bull's-eye.

• Horizontal Motion: This is consistent and unaffected by gravity. If a bullet is fired from a rifle parallel to the ground, its horizontal velocity remains constant. This is because there is no horizontal force acting on it (assuming air resistance is negligible).
• Vertical Motion: In contrast, vertical motion is greatly influenced by gravity. As the bullet travels forward, it also falls downward at an accelerating rate due to gravity.

The bullet's path is a combination of these two motions. While it moves steadily towards the target horizontally, it simultaneously drops towards the ground. This combined motion forms a curved path, which is a hallmark of projectile paths.

Time of flight is the duration an object remains in motion from the moment it is projected until it hits the target. In the given exercise, understanding time of flight helps explain why doubling the distance results in the bullet falling four times as far. For horizontal motion, time of flight is crucial in determining how long the bullet travels towards the target at a given constant speed. As calculated in the original problem, if the target's distance is doubled, the time of horizontal flight doubles (\(t_2 = 2t_1\)), simply because the bullet covers more horizontal space.Vertically, the longer the time an object is in the air, the more it is affected by gravity. Therefore, if the time of flight increases, the gravitational pull causes a greater drop. This directly explains the increased vertical distance fallen (\( H_B = 4H_A \)) when the shooting distance doubles. Thus, understanding time of flight explains how variations in distance impact both the horizontal and vertical travel, giving a comprehensive picture of a projectile's motion.