Unsolved Mysteries yields a kinematics problem

The opening story in the series yields a mystery about a man having fallen to his death. An investigator states that he felt uncomfortable with a distance traversed (on horizontal plane) relative to the vertical height fallen. The height given for the building is approximately 10 stories. The distance traveled on the relative horizontal plane axis from such falling height is approximately 45 feet.

Actually neglecting wind conditions (e.g. air resistance) if it is entirely possible that the man could have traveled the distance described.

For conventional axis of motion the problem, it could be a three dimensional problem. However, we could simplify this to a simple two dimensional problem. Namely xz or yz relative planes.

Kinematic motion, if you are familiar, can be described by component axis motion.

In this problem, first we’d solve the time elapsed to impact.

For the vertical falling part of this problem we are working in imperial units.

So gravity is given typically by 32.17405 ft/s²

We are missing one detail namely the height at impact site. We could do some estimation. I used an estimation of approximately between two and three stories, or approximately 25 ft.

So the falling distance traveled, I estimated at around 75 ft.

The kinematic equation of motion then is

75ft = 1/2*(32.17405 ft/s²)*t²

or solving for t we have,

t = 2.159199462 seconds

That is it there was around 2.16 seconds of falling time before impact.

Now we can resolve the motion along the horizontal axis. In this case, after jumping, there assumed to be no added acceleration (i.e., neglecting air resistance or additional forces along the horizontal axis of motion).

Thus the kinematic problem becomes how fast was the man traveling at the point of jumping off the building.

We know the distance traveled is 45 ft.

The kinematic equation by given knowns then is

45 ft = v (2.159199462 sec)

or

v = 20.84105743 ft/s

How reasonable is this initial velocity relative to average top sprinting speeds?

Let’s convert the units to miles per hour

Using the conversion calculator (https://www.google.com/search?q=ft%2Fs+to+mph&rlz=1C1CHBF_enUS864US865&oq=ft%2Fs+to+mph&aqs=chrome..69i57j6j0l2j69i58.3049j0j7&sourceid=chrome&ie=UTF-8)

The man could have been estimated possible traveling

14.209811884091 miles per hour

The average top human sprint is approximately 15 miles per hour

Does this mean that the man had enough distance to accelerate to the necessary speed when jumping to land at the point of impact? It is hard to say if the building provided requisite conditions to reach such sprinting speed (in terms of obstacles and clearance). It is also hard to say without knowing what his top sprinting speed was relative to averages, but we can say that motion described was not likely extraordinary either. Meaning he probably wasn’t dropped out of a helicopter or having additional circumstances required at least relative to the conditions described. That is provided some additional assumptions hold true. Namely, lacking negative wind conditions unfavorable to distance traveled, and given that approximate falling height is correct. If the falling height decreases, the sprinting speed required increases, while with increasing falling height means the running speed requirements decreases.