## Why paths bend A particle’s path bends because a **uniform vertical field** keeps changing its **vertical velocity** moment by moment. The horizontal part of the launch keeps carrying the particle sideways, but the field continuously pushes its motion upward or downward. That means the path is not just set at launch — it is reshaped throughout the flight. A useful intuition is to imagine two identical launches, but with the “background push” flipped. In a downward field, the path bows downward; in an upward field, it bows upward. **The launch starts the motion, but the field edits it over time.** <viz id="0"></viz> **Toggle the field direction** and **replay the motion**. If both views are visible, **compare the two paths** carefully. Watch the acceleration arrow stay fixed vertically while the trajectory gradually curves in that direction. Notice the key asymmetry: nothing in the horizontal direction is forcing the particle to speed up or slow down, so the sideways motion stays steady. The curvature comes entirely from the vertical field. That is the core idea behind projectile-style motion in a vertical force field. Once that clicks, you can start asking a more interesting question: if the field is fixed, how much can the **launch angle** change the shape of the path? That is the next layer of intuition in <ref slide="2">Angle and field explorer</ref>. ## Angle and field explorer The **launch angle** determines how the initial speed is split between sideways motion and vertical motion. A shallow launch spends more of its speed moving horizontally; a steep launch spends more of it climbing or diving vertically. Then the **field direction** either works against that initial vertical motion or strengthens it. This is why the same speed can produce very different paths. You are really combining two ingredients: the direction you launch, and the direction the field keeps pushing afterward. **The trajectory is the result of that combination, not either piece alone.** <viz id="1"></viz> **Drag the angle slider** through shallow and steep launches, then **flip the field direction**. Watch how the velocity vector starts in the chosen direction, while the path gradually bends with the field. Look for whether the particle reaches a highest point, a lowest point, or keeps moving away. A nice pattern appears here: with a downward field, an upward launch can rise at first and later turn downward. With an upward field, that same launch can keep bending upward more strongly instead. So the turning behavior depends on whether the initial vertical motion and the field are competing or cooperating. This also explains why angle alone is not enough to predict the motion. To understand the path cleanly, it helps to split the launch into **horizontal and vertical components** — the two pieces that evolve differently. That is exactly what you will see in <ref slide="3">Component breakdown</ref>. ## Component breakdown A velocity vector becomes much easier to reason about when you split it into **components**. The **horizontal component** tells you how fast the particle moves sideways. The **vertical component** tells you how fast it initially moves up or down. In a uniform vertical field, these two components do not behave the same way. The big idea is simple but powerful: **the field only acts vertically**. So horizontal motion keeps its initial pace, while vertical motion is the part that gets steadily modified. This is the same kind of decomposition mindset that shows up all over technical work: break one complicated quantity into independent directions, then track what changes in each one. <viz id="2"></viz> **Drag the launch vector** to change both its direction and length, and **flip the field direction**. Compare the component arrows with the later path. Notice that the sideways component sets how quickly the particle moves across, while the vertical component is the one the field keeps changing. This decomposition gives you a cleaner mental model of curvature. If sideways motion stays steady but vertical motion keeps being nudged, the path cannot remain straight. It must bend because one component is constant while the other evolves. That leads naturally to a time-based view: what exactly happens to each component as seconds pass? In <ref slide="4">Field through time</ref>, you will watch the vertical component change continuously and see the turning point emerge when it reaches zero. ## Field through time A vertical field does not change the motion all at once. It changes it **continuously through time**. At each moment, the particle’s **vertical velocity** is a little larger or smaller depending on the field direction, while the horizontal velocity stays unchanged. This produces one of the most important insights in the lesson: a turning point happens when the vertical component passes through `0`. Before that moment, the particle is still moving upward if the vertical velocity is positive. After that moment, it is moving downward if the vertical velocity becomes negative — or the reverse if the field points upward and the particle started downward. <viz id="3"></viz> **Scrub through time** slowly and **flip the field direction**. Watch the horizontal velocity stay steady while the vertical velocity changes step by step. **Pause at the instant the vertical component reaches `0`** and observe that this is exactly where the path changes from rising to falling, or vice versa. This is the mathematical tipping point for the motion: not when the particle is “about to turn” in a vague sense, but precisely when its vertical speed has been driven to zero. The path’s highest or lowest point is a direct consequence of that sign change. Once you can read the motion through time, you are ready to compare two worlds side by side: same launch, opposite fields. That makes the role of field sign impossible to miss in <ref slide="5">Field sign comparison</ref>. ## Field sign comparison If you keep the **initial speed** and **launch angle** fixed but reverse only the field sign, you isolate the effect of the environment from the effect of the launch. This is a powerful comparison because it removes distractions: the particle starts the same way in both cases, yet the later motion can look qualitatively different. One field can create a high point and bring the particle back down, while the opposite field can create a low point or keep pushing the particle farther upward. **Changing only the sign of the field is enough to reverse the bending behavior.** <viz id="4"></viz> **Adjust the launch speed and angle**, then **compare both panels**. Look at where each path bends, whether a turning point appears above or below the start, and how long the particle keeps traveling before reaching the boundary or returning. This side-by-side view reinforces a deep idea: trajectory shape is not just about how hard you launch. It is about the interaction between the initial velocity and the persistent field. Reversing the field reverses the long-term trend in vertical motion, even though the launch itself is unchanged. At this point, you have all the ingredients — launch direction, velocity components, and time evolution. The final step is to compress them into one unified picture in <ref slide="6">Putting it together</ref>. ## Putting it together You can now think of particle motion in a vertical field as a compact rule set. Start with the **initial velocity**, split it into **horizontal** and **vertical components**, then let the **field** steadily modify only the vertical one. From that alone, you can predict the path’s curvature, whether a turning point appears, and how reversing the field changes the story. The most important intuition is that the particle is always doing two things at once: moving sideways at a steady rate and being continuously pushed vertically. **Curved motion is what those two simultaneous behaviors look like when combined.** <viz id="5"></viz> **Flip the field direction** and **trace the path with your eyes** from launch to turning point to later motion. Then **compare the velocity components** with the final curve: the horizontal part carries the particle across, and the changing vertical part determines how the path bends. If you leave with one mental model, make it this one: the launch sets the starting motion, but the field writes the ongoing vertical correction. That is why identical launches can diverge dramatically when the field sign changes. With that picture in place, you are no longer memorizing trajectory shapes. You are reading them from the underlying component-wise dynamics, which is the more general and more useful way to think about motion in fields.