Projectile motion involves the movement of an object under the influence of gravity, typically characterized by a parabolic trajectory. To analyze such motion, especially in cases of horizontal launches, a systematic approach can be employed. This involves breaking down the motion into horizontal (x-axis) and vertical (y-axis) components.

When a ball rolls off a table, the first step is to sketch the trajectory on the x and y axes. The ball will follow a curved path due to gravity, and it is essential to identify key points, such as the initial position (point A) and the final position (point B) when the ball hits the ground. The next step is to determine the target variable, which in this case is the time (t) it takes for the ball to reach the ground.

To find the time of flight, we can use the equation for horizontal motion:

$$\Delta x = v_{x} \cdot t_{AB}$$

where $$\Delta x$$ is the horizontal displacement, $$v_{x}$$ is the horizontal velocity, and $$t_{AB}$$ is the time from point A to point B. In horizontal launches, the initial velocity is entirely in the x-direction, and the vertical component of the initial velocity is zero.

For the vertical motion, we can use the following kinematic equation, which relates displacement, initial velocity, acceleration, and time:

$$\Delta y = v_{yA} \cdot t_{AB} + \frac{1}{2} a_{y} \cdot t_{AB}^2$$

In this equation, $$\Delta y$$ is the vertical displacement (which is negative if the object falls), $$v_{yA}$$ is the initial vertical velocity (zero for horizontal launches), and $$a_{y}$$ is the acceleration due to gravity (approximately -9.8 m/s²).

For example, if the table height is 2 meters, the vertical displacement is -2 meters. Plugging in the known values allows us to solve for time:

$$-2 = 0 \cdot t_{AB} + \frac{1}{2} (-9.8) \cdot t_{AB}^2$$

Solving this gives us the time of flight, which is approximately 0.64 seconds.

Once the time is determined, we can find the horizontal displacement using the previously mentioned equation. If the horizontal velocity is 3 m/s, the horizontal displacement can be calculated as:

$$\Delta x = 3 \cdot 0.64 = 1.92 \text{ meters}$$

In summary, for horizontal projectile motion, the initial velocity is entirely in the x-direction, and the vertical motion is influenced solely by gravity. The time of flight can be calculated using vertical motion equations, while horizontal displacement can be determined using the constant horizontal velocity. Understanding these principles allows for effective problem-solving in projectile motion scenarios.