In studying how objects move through the air, it’s important to understand the two main directions they go: side to side (horizontal) and up and down (vertical). These two parts act separately because of gravity. Knowing how they work together helps us see the whole path of a flying object.

Horizontal Motion

The side-to-side motion is straightforward. When nothing pushes or pulls an object sideways (like air resistance), it moves at a steady speed. This means it keeps going further without slowing down. We can use this simple math to find out how far it went:

$x = v_{0x} \cdot t$

In this formula, $v_{0x}$ is the starting speed going sideways, and $t$ is how long it's been flying. The cool thing is that this side motion doesn’t change because of gravity. So, it moves in a straight line until it combines with the up-and-down motion.

Vertical Motion

On the other hand, the up-and-down movement is affected by gravity. This force, which pulls things down, is often about $9.81 \, \text{m/s}^2$. Because of this, the object speeds up as it goes down and slows down as it goes up. We can use this equation to describe where it is vertically over time:

$y = v_{0y} \cdot t - \frac{1}{2} g t^2$

Here, $v_{0y}$ is the starting speed going upward. When the object first launches (if $v_{0y}$ is positive), it goes up, stops for a moment at the top, and then comes back down because of gravity.

Putting It All Together

When we combine both horizontal and vertical motions, we get the well-known curved path, or parabolic trajectory, that objects follow when they’re thrown.

To sum it up: the sideways motion stays the same speed, but the up-and-down motion changes because of gravity, making it go faster or slower as it rises and falls. Knowing how these parts work helps us figure out exactly where a projectile will go, how far it will land, and its highest point. This is very important in understanding how things move.