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Projectile motion is a type of motion experienced by an object thrown into the air and influenced by gravity only. This motion has two components: horizontal and vertical motion. These two components are independent of each other. The horizontal motion occurs with constant velocity, while the vertical motion is subject to the acceleration due to gravity. Both the vertical and horizontal motions affect where the projectile will land.

In the given exercise, the ball rolling off the table is an example of projectile motion. Once the ball leaves the table, it becomes a projectile. The horizontal distance it travels depends on its initial horizontal velocity and the time it is in the air. On Earth, this time of flight is determined by the gravitational pull of 9.8 m/s². The ball's motion on another planet with different gravity, such as Planet X in this exercise, will result in a different horizontal distance.

The concept of horizontal velocity is crucial in understanding projectile motion. Horizontal velocity is the speed of an object moving parallel to the horizontal surface. Importantly, once an object is in motion, no horizontal acceleration affects it under the assumption of minimal air resistance. Therefore, this velocity remains constant throughout its flight.

In the scenario provided, the ball maintains the same initial horizontal velocity both on Earth and Planet X. This consistent speed allows us to calculate distances traveled across different gravitational fields, assuming no other forces are acting on the ball. The ball travels 2.76 times further on Planet X than on Earth because the gravity on Planet X influences the time of flight, not the horizontal velocity.

The time an object spends in the air is termed 'time of flight.' It's a vital component in calculating projectile motion. The time of flight is influenced by the object's initial vertical velocity and the acceleration due to gravity. For objects released horizontally, like the ball in our exercise, the time of flight only depends on the initial vertical drop and gravitational acceleration.

On Earth, the time of flight is derived from the equation: \(t = \sqrt{\frac{2h}{g}}\). This is because the gravitational force pulls the ball to the ground. When calculating the time of flight on Planet X, the difference in gravitational acceleration changes the time the ball spends in the air. With reduced gravity, the ball remains airborne for a longer period, thus covering a greater horizontal distance.

Different planets exert different gravitational forces on objects, a factor that significantly influences projectile motion. Earth has a standard gravitational acceleration of approximately 9.8 m/s². However, this value is not universal and varies with different celestial bodies.

In our example, Planet X has weaker gravitational pull compared to Earth. This lessened gravity means that objects will fall more slowly, extending their time of flight when thrown horizontally. By knowing the relative distance the ball travels on both surfaces (i.e., Earth and Planet X), and using the formula \(g_x = \frac{g}{(2.76)^2}\), we affirm that Planet X's gravitational force is approximately 1.28 m/s². This significant difference helps compare and understand planetary masses and sizes, providing insights into similar or more diverse extraterrestrial environments.