Question

current attempt in progress
the figure shows three paths for a football kicked from ground level.
in the following questions, you will be ranking these paths. if multiple paths rank equally, use the same rank for each, then exclude the intermediate ranking (i.e. if objects a, b, and c must be ranked, and a and b must both be ranked first, the ranking would be a:greatest, b:greatest, c:third greatest). if all paths rank equally, rank each as greatest.
ignoring the effects of air, rank the paths according to time of flight.
path 1
path 2
path 3
ignoring the effects of air, rank the paths according to initial horizontal velocity component.

Explanation:

Step1: Relate flight time to max height

For projectile motion starting/ending at ground level, time of flight depends on maximum height ($H$):
$$t = 2\sqrt{\frac{2H}{g}}$$
Greater $H$ means longer flight time. Paths 1 and 2 have equal max height, Path 3 has the highest.

Step2: Rank flight time

Path 3 has longest flight, Paths 1 and 2 are equal and shorter.

Step3: Relate horizontal velocity to range

Horizontal velocity $v_x = \frac{R}{t}$, where $R$ is horizontal range. Path 3 has largest $R$ and longest $t$; Path 1 has smallest $R$ and shortest $t$.

Step4: Rank horizontal velocity

Larger $R/t$ means greater $v_x$. Path 3 > Path 2 > Path 1.

Answer:

Ranking by time of flight:

Path 1: Second greatest
Path 2: Second greatest
Path 3: Greatest

Ranking by initial horizontal velocity component:

Path 1: Third greatest
Path 2: Second greatest
Path 3: Greatest