A horizontal launch begins with uy=0, not zero vertical acceleration. With downward positive and drag neglected, h= ½gt2 gives t=√g2h. Horizontal motion gives x= uxt. Both components use the same time. At impact, vy= gt and speed is √ux2+vy2.
Drag opposes velocity relative to the surrounding fluid. It generally grows with speed, frontal area and fluid density, with shape also important. It can reduce horizontal speed and prevent an ideal parabolic path. In a vertical fall, terminal velocity occurs when upward drag equals weight: resultant force and acceleration are zero, while velocity remains nonzero.
Example 1
A ball is launched horizontally at 8.00ms−1 from 4.90m. Use g=9.80ms−2 and ignore drag.
t=√2×9.804.90=1.00s. Range =8.00×1.00=8.00m.
Example 2
The same ball lands with vx=8.00 and vy=9.80ms−1 downward. Find speed.
v=√8.002+9.802=12.7ms−1. It is directed arctan(8.009.80)=50.8∘ below horizontal.
Watch out: Without drag, equal initial vertical velocities and heights give equal fall times, regardless of horizontal speed.