projectile motion
Projectile motion is the arcing flight of anything you throw, kick, or launch and then let go — a basketball's swish, a thrown ball, a jet of water from a hose. Once it leaves your hand (and if we ignore air resistance), the only force on it is gravity, so it follows a smooth curved path back to the ground. The big idea, and it is a genuinely clever one, is that this curved motion is really two simple motions happening at once.
Precisely, projectile motion is free fall in two dimensions, and its secret is that the horizontal and vertical motions are independent. Horizontally there is no force (ignoring air), so the horizontal velocity stays constant and the horizontal position grows steadily: x = v_0x t. Vertically there is gravity, so the vertical motion is ordinary free fall with acceleration g downward: v_y = v_0y - g t and y = v_0y t - (1/2) g t^2. You solve each direction on its own with the constant-acceleration equations and combine the results.
This independence is why a bullet fired horizontally and a bullet simply dropped from the same height hit the ground at the same instant: their vertical motions are identical, and the horizontal speed of the fired bullet does not delay its fall at all. The idealization to be honest about is 'ignore air resistance' — for a thrown ball over short ranges this is a fine approximation, but for a fast, light, or long-flying object (a golf ball, an arrow), air noticeably shortens and bends the real path away from the ideal one.
Roll a ball off a table 1.25 m high with a horizontal speed of 3 m/s. The fall takes t = sqrt(2 x 1.25 / 9.8) = 0.5 s, set entirely by the vertical drop. In that time it travels x = 3 x 0.5 = 1.5 m horizontally — and it lands at the same instant as a ball simply dropped from the table edge.
Horizontal and vertical motions are independent: the sideways speed does not change the fall time.
The horizontal velocity does not affect how long the object stays in the air. Time aloft depends only on the vertical motion (launch height and vertical launch speed). Faster sideways just means it lands farther away, not later.