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Horizontal velocity describes how fast an object travels along the horizontal axis. In projectile motion, it's essential because it helps determine how far an object will travel horizontally before landing. Imagine someone sliding horizontally off the end of a water slide. Their initial horizontal velocity is dictated by how fast they exit the slide.

Given a person hits the water at a horizontal distance of 5.00 meters in 0.500 seconds, we can calculate their horizontal velocity using the formula:

• \(\text{distance} = \text{velocity} \times \text{time} \)

Plugging in the numbers, we get:

• \(5.00 = v_x \times 0.500\)

This simplifies to a horizontal velocity \(v_x\) of 10 meters per second.

Once calculated, this velocity remains constant throughout the motion if air resistance is ignored, meaning we'll use the same value to determine how far the swimmer travels horizontally until they splash into the water.

Vertical motion deals with the movement of an object along the vertical axis, which is influenced mainly by gravity. For a swimmer sliding off a water slide, the vertical motion refers to how the swimmer falls due to gravity after leaving the slide.

When an object is in free fall, the height of the fall can be determined using the formula for vertical motion determined by gravity:

• \(H = \frac{1}{2} g t^2\)

The term \(g\) represents gravity (approximately \(9.81 \text{m/s}^2\) on Earth), and \(t\) is the time taken, in this case, 0.500 seconds.

Substituting the known values into the formula provides a measure of how far the swimmer drops from the slide height to the water surface. After performing the calculation, we find that the fall height or \(H\) is approximately 1.23 meters. This calculated height is the vertical distance over which the swimmer moves under the influence of gravity alone.

Gravity is an ever-present force pulling objects toward the center of the Earth. In projectile motion, like our sliding swimmer, gravity is the force responsible for their acceleration in the vertical direction.

On Earth, gravity provides a constant acceleration of \(9.81 \text{m/s}^2\), affecting all objects in free fall equally. In the case of the swimmer, once they leave the water slide and begin to descend, gravity dictates their vertical motion. Despite moving forward horizontally, the swimmer's downward motion accelerates steadily due to gravitational attraction.

This consistent force allows us to calculate different parameters of motion, including how long it will take the swimmer to reach the water and how high the end of the slide is from the water's surface. Understanding the role of gravity in vertical motion provides clarity on why certain equations, like \(H = \frac{1}{2} g t^2\), are used to predict distances in projectile motion scenarios.