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.

your answer is partially correct.

ignoring the effects of air, rank the paths according to time of flight.

path 1 third greatest
path 2 third greatest
path 3 greatest

Explanation:

Step1: Relate flight time to max height

For projectile motion (launched and landing at same height), time of flight depends on vertical motion. The time to reach max height is $t_{up}=\frac{v_y}{g}$, and total flight time is $t_{total}=2t_{up}=\frac{2v_y}{g}$. Max height $H=\frac{v_y^2}{2g}$, so greater $H$ means greater $v_y$, thus greater flight time.

Step2: Compare max heights of paths

From the figure, Path 3 has the greatest max height, Paths 1 and 2 have equal lower max heights.

Step3: Rank flight times

Paths with equal max height have equal flight time. So Path 3 has the longest flight time, Paths 1 and 2 have equal shorter flight times.

Answer:

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