Read and reason
Solve the landing-height condition
To find when a projectile reaches flat ground at y=0, solve 0=h+vy0t−(1/2)gt². For nonnegative height and a suitable launch, use the nonnegative root t=(vy0+sqrt(vy0²+2gh))/g. This formula combines the upward initial velocity with gravitational acceleration. Units inside the square root are (m/s)² and the final result is seconds.
Distance depends on horizontal speed and flight time
With horizontal acceleration zero, x displacement equals vx0×t_flight. If you double vx0 while keeping vy0 and launch height fixed, impact time is unchanged and range doubles. Changing the angle at fixed speed changes both components, so the range result is not generally monotonic in angle. Starting height also matters: from a platform, a 45° launch is not automatically maximum range.
Interpreting the apex
If vy0>0, the model reaches vy(t)=0 at t_apex=vy0/g. The apex height is h+vy0²/(2g). If the launch is horizontal (vy0=0), the launch point is already the maximum altitude in this model. The point t_apex is not the impact time unless the trajectory has some special boundary; compute impact from y(t)=0 separately.
Check reasonableness before accepting a solution
The negative algebraic root normally corresponds to a time before the modeled launch and is rejected. Check whether y(t) is nonnegative before impact and whether a trial duration includes impact. A plotted trajectory represents position against position—not against time—so axes must be explicitly labeled x (m) and y (m). A ballistic classroom model is not an experimental observation or a real launch plan.