The trajectories below represent the motion of a ball during after three separate kicks. Across all three kicks, the ball reaches the same vertical height h.

During which of the kicks was the ball's initial horizontal velocity largest?

Question

The trajectories below represent the motion of a ball during after three separate kicks. Across all three kicks, the ball reaches the same vertical height h.

During which of the  kicks was the ball's initial horizontal velocity largest?

BlackTom AI · Accepted Answer

To compare the kicks, start from the fact that the maximum height reached by a projectile is determined by its initial vertical speed, not its horizontal speed. Since all three trajectories reach the same height h, the initial vertical component vy0 must be the same for all three kicks. In projectile motion, the maximum height is h = vy0^2 / (2g), so identical h implies identical vy0 across Kick 1, Kick 2, and Kick 3.

With vy0 the same and assuming launches from the same initial height, the time of flight is determined by the vertical motion: T = 2 vy0 / g for a symmetrical trajectory landing back at the same level. Because vy0 is the same for all kicks, the time of flight T is also the same for all three trajectories.

The horizontal displacement, or range, of a projectile is R = vx * T. Since T is the same for all three kicks, the only factor that changes the range is the initial horizontal velocity vx. The kick with the largest vx will travel the farthest horizontally and thus have the widest trajectory on the horizontal axis.

Option analyses:
- All the same: This would be true only if vx were identical for all kicks as well, but the problem’s figure (and the question prompting which kick has the largest horizontal velocity) implies differing vx values; with vy0 identical, differing vx leads to different ranges. Therefore this option is incorrect.
- Kick 1: If Kick 1 had the smallest vx, it would have the smallest range, not the largest; this option is inconsistent with the question asking which kick has the largest initial horizontal velocity.
- Kick 2: Similar reasoning as Kick 1; unless Kick 2’s vx is the largest, this option would be false. Since we are identifying the largest vx, Kick 2 would only be correct if it shows the longest horizontal extent, which the prompt indicates is Kick 3.
- Kick 3: This is the option indicating the largest horizontal velocity. Given vy0 is the same for all kicks and time of flight is the same, the trajectory with the greatest vx will have the greatest horizontal reach, corresponding to the largest initial horizontal velocity.

In summary, equal maximum heights force equal vy0 and equal flight time, so the only way to differentiate the kicks is by vx. The kick with the largest horizontal velocity yields the longest horizontal extent, which corresponds to Kick 3.

The trajectories below represent the motion of a ball during after three separate kicks. Across all three kicks, the ball reaches the same vertical height h. During which of the kicks was the ball's initial horizontal velocity largest?Single choice

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