One ball, two component stories and a shared clock — Physics & Environmental Science

A ball leaves a hand with both forward and upward velocity. In an ideal near-Earth model without drag, gravity supplies downward acceleration throughout the flight. Horizontal velocity stays constant while vertical velocity decreases and later becomes negative. At the top the ball still moves forward; zero vertical velocity is not zero total velocity or zero acceleration.

A question about maximum height uses a different event condition from a question about landing. Set vertical velocity to zero for the apex; set vertical position to the actual landing height for impact. An equation can have two mathematical roots corresponding to launch and a later return. Select the event the question asks for, then use the same time in the horizontal equation.

Graphs provide independent checks. Position slopes describe velocity; velocity slopes describe acceleration and signed velocity areas describe displacement. Split an area at a reversal to find distance. A real thrown ball can depart from an ideal parabola because of drag, spin, launch uncertainty or landing geometry. Report those limits rather than stretching an elegant equation beyond its assumptions.