Skateboard Mystery
solution 
  Copyright ©2020 by Paul Niquette. All rights reserved.


Skateboarding is an action sport enjoyed by more than 11 million skateboarders worldwide and will be represented in the 2020 Summer Olympics. The Skateboard Mystery puzzle is based on an analysis of Aaron Homoki's record-setting jump using the law of conservation of energy as reprised in this sketch...

jump dimensions

...wherein one observes an apparent -- and unlawful -- destruction of energy immediately after the skateboard contacts the ground.  The amount 'destroyed' is equal to the potential energy at zenith m g (yS + yJ)...
Where did all that energy go?
One might expect that some of the potential energy would have been converted somehow on impact to increase the kinetic energy = (m/2) (vX + ∆vX)2.  However, vX appears to remain quite constant in the video. 
If anything, vX may actually decrease at the completion of the jump, inasmuch as, with its wheels temporarily indented, the skateboard would be briefly 'braked', tending to throw the skater forward.
Of course, sound energy is released by the impact of the wheels on the pavement, but whether that would  account for the loss of potential energy seems quite doubtful. 
The likely explanation is that the skater absorbs most of the energy in eccentric muscular contraction.  (Nota bene, Homoki expended plenty of concentric muscular contraction while [a] climbing those 25 stairs carrying his skateboard, [b] accelerating to vX, then [c] jumping upward in the launch.)  Now, the subject of how skeletal muscles work is fascinating but beyond the scope of the puzzle. 
Sophisticated solvers will observe that the horizontal velocity vX acts in reciprocity with the required height of the skater's jump yJ: Faster the skateboard approaches the launch: the lower the required height at the zenith.   The relationship can be expressed in this general equation for any skateboard stairway jump...
vX = xS / [2 (yJ + yS) / g]1/2
Homoki's record-breaking jump of the 25-step stairway is analyzed in the graph below.  It is based on the assumption that the height of his jump yJ = 1 meter (39.4 inches) above the top stairstep.  The duration of the jump is calculated to be 1.5 seconds, and thus the required horizontal velocity vX = 4.45 m/s (10 mph).

jump analysis

Let us suppose Homoki sets about to establish a horizontal speed higher than vX = 4.45 m/s prior to the jump.  There is an advantage in doing so.  We observed above that the amount of potential energy to be absorbed in the impact at ground level is proportional to the height of the trajectory's zenith above ground level (yS + yJ).  
Accordingly, a 10% increase in speed to vX = 4.9 m/s (11 mph) will decrease the duration of the jump by 10% to 1.35 seconds and allow the height of the jump to decrease by 43% so that yJ  = 0.57 meters (22.4 inches).  The zenith height will be decreased by 7.8% such that yS + yJ = 5.1 meters.
Solvers may ask, "What horizontal speed vX would allow Homoki to clear the stairway with no jump (yJ = 0)?"  The answer is vX = 7.0 m/s (15.7 mph), which compares to running a 100-meter dash in a leisurely 14.2 seconds.
Thus we have a solution to the Skateboard Mystery.   Meanwhile, our analysis has produced another Skateboard Mystery that enjoys special significance for Puzzles with a Purpose.  Let us consider the practical subject of...
Skateboard Safety
The video referenced in the Skateboard Mystery puzzle, is merely one of many available on the worldwide web. Unfortunately some show skateboarding accidents, and the injury statistics are grim.  Indeed Aaron Homoki was hospitalized during his repeated attempts to establish the 25-stair record.

Here is a proposed solution for consideration by skateboarding enthusiasts who are determined to set records by interacting with high objects -- particularly jumping from the tops of stairways.  It calls for analysis and planning and may be as effective as the safety hardware skateboarders might own but -- ugh -- often decline to wear.
jump safety
 
The real Skateboard Mystery: Why would any record-seeking skateboarder not...
  • find a suitable outdoor stairway or curb for safely practicing jumps,
  • find a measuring tape, a piece of chalk, a piece of ribbon, a couple of posts, and a stopwatch,
  • measure xS' then visit the site of the objective stairway to determine xS and yS,
  • tie the ribbon at variable heights on the posts to determine skater's maximum value for yJ,
  • use trajectory equations to calculate xJ,  xS', and minimum value for vX,
  • use stopwatch measurements between chalk lines to confirm skater's achievable vX,
  • use chalk lines to mark the launch-point and xS',
  • place the posts and ribbon at xJ and yJ respectively, then...
...practice, practice, practice?


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