projectile motion explained

Drag the time t (seconds) with the slider and both balls move together. The blue ball on the left was simply released and the gold ball on the right was thrown horizontally from the same height; the faint gold dashed line is the path the thrown ball will follow. The blue dashed line joining the two balls always stays level, which means that at every instant they are at exactly the same height. However far the thrown ball travels sideways, it falls exactly as much as the dropped one, so the two land at the same instant. Gravity pulls only downward and does not care how fast the ball moves sideways, so horizontal motion has no way to interfere with the vertical fall; that independence is the key to projectile motion.

Now take the horizontal direction on its own. Ignoring air resistance, nothing pushes or holds the ball horizontally, so its acceleration is 0 and its horizontal velocity never changes. Drag the horizontal-velocity slider vx and the gold marks taken at equal time intervals spread wider or closer, but in every case the spacing stays even and the blue velocity arrows all have the same length. So the horizontal position is x = vx·t, the simplest motion there is, in direct proportion to time. The horizontal direction of a parabola runs at one steady speed from start to finish.

Next, the vertical direction alone. Even with an initial upward speed, gravity keeps pulling down, so the vertical motion is exactly the constant-acceleration motion of the last lesson; taking up as positive, the acceleration is always −g, pointing down. Drag the time t and the gold ball rises fast at first, slows, pauses for an instant at the top, and comes back down, while the orange arrow beside it shows the direction and size of the vertical velocity. The short ticks on the left mark the ball's height at equal time intervals: crowded near the top and spread out lower down, so the way up mirrors the way down. The height is y = vy·t - ½g t², the constant-acceleration formula unchanged. The top is the moment the vertical velocity reaches 0, and the gold dashed line H marks that maximum height; forget the horizontal and this is just a ball thrown straight up.

Now combine the two motions. At every instant the horizontal position is x = vx·t, advancing to the right in even steps, and the vertical position follows y = vy·t - ½g t², up and back down. Drag the time t and the gold ball flies along the faint curve, which is the parabola. The blue dot on the ground and the orange dot on the left axis are shadows that track the horizontal position and the height separately, and the ball always sits where the dashed lines from the two shadows meet. Curved as it looks, nothing new is going on: uniform horizontal motion and uniformly accelerated vertical motion simply run on the same clock.

Finally, change the launch angle. Keep the launch speed fixed and turn only the angle θ with the slider, and a ball thrown just as hard goes very different distances and heights. Raise the angle and it soars but does not go far; lower it and it skims low and again falls short. The longest range comes exactly in the middle, at 45 degrees, and the faint dashed curve is that trajectory. Two angles equally far from 45 degrees, such as 30 and 60 degrees, give the same range: one flies low and fast, the other high and leisurely, and both land in the same spot. The blue tick on the ground is the range and the orange dot the maximum height, which keeps growing as the angle steepens and is greatest for a throw straight up at 90 degrees. The range is given by R = v0² sin2θg.