Would I be able to jump noticeably higher at the top of a skyscraper because there is less gravity? At the top of the Burj Khalifa?
Does anyone have a video or something of someone testing this? I was scouring Google for answers but haven't found any.
No, not really. The acceleration of gravity is inversely proportional to the distance from the center of the Earth so that:
$\large\frac{g_{Burj\ Khalifa}}{g_{surface}}=\left(\frac{r_{surface}}{r_{Burj\ Khalifa}}\right)^2=\left(\frac{6400\ km}{6400.83\ km}\right)^2=0.9997$
where $r_{Burj\ Khalifa}$ is the radius of the Earth (6400 km) plus the height of the building (0.83 km). Since gravity is 99.97% the strength at the top of the building that it is on the surface of the Earth, you would be unlikely to measure any difference in jump height.
As far as it concerns the gravitational effect in a jump at high altitude, as you mentioned in your question,i am fully covered by the previous answers.
There's though a significant effect due to the atmospheric air density at high altitudes .
I am copying from wikipedia:
'since the thinner atmosphere of higher altitude provides less air resistance, locations such as Mexico City and Sestriere have previously been the sites of records in the sprint and jump events'
source:https://en.wikipedia.org/wiki/List_of_world_records_in_athletics
Not jump higher in any easily measurably manner. I would very much doubt that any video exists of it because the effect is so small. But if you want to analyse it more carefully, you would have to take into account that because your velocity would be faster at that height, so your air friction induced drag would increase as compared to ground level, so you may not get any higher.
