Thread by @anderssandberg

#FridayPhysicsFun – I am back home in the apartment where I grew up on the 11th floor. That is about 30 m down to the street, and as a kid I often considered the fate of toys dropped from the balcony. How does falling really work? en.wikipedia.org/wiki/Hagalund

The schoolbook answer is that the gravitational force F=mg accelerates the object as per Newton’s second law of motion F=ma and the falling object has an acceleration a=g because the mass factor cancels from both equations.

The velocity becomes v(t)=gt at time t, and the distance travelled d(t)=(1/2)gt^2. I remember kid-me inverting the later formula to t=sqrt(2h/g) and checking by dropping marbles that they took about 2.47 s to hit the ground. Fortunately nobody got hurt.

This ignores air resistance, which gives a velocity dependent force proportional to the square velocity, the air density rho, the area A, and a complicated drag coefficient C. Adding it to the equation makes things complicated (kid-me was annoyed!) en.wikipedia.org/wiki/Drag_(phy…

The basic effect is that velocity doesn’t keep on increasing but reaches terminal velocity sqrt(2mg/rho A C). A parachute boosts A and C, while a heavy object will fall faster in air (despite Galileo!)

A nice rule of thumb is that objects with density close to water (like humans and rain) have terminal velocity 90 m/s times sqrt(size in metres). For humans this is 70 m/s, insects and rain about 9 m/s or less.

Really small objects have viscosity-dominated drag and slow more since the drag is proportional to velocity and not the squared velocity, which matters more at low velocity. en.wikipedia.org/wiki/Stokes%27…

Assuming constant gravity is wrong. Up here the gravitational acceleration is about 9.4 millionths weaker than at the ground since I am closer to space.

Inside Earth gravity declines as one approaches the core. A classic physics problem is to calculate the time it would take to fall through a hole straight through Earth. (It also shows up in Gregory Benford’s SF novel “Tides of Light”) hyperphysics.phy-astr.gsu.edu/hbase/Mechanic…

Assuming Earth has constant density is not a good approximation. Gravity is roughly constant in the mantle, increasing slightly as the core is approached and then declines. This produces a slightly different and messier answer. en.wikipedia.org/wiki/File:Eart…
aapt.scitation.org/doi/10.1119/1.…

There is also a centrifugal acceleration that is 1.7 thousands of the gravitational. Since I am about at 60 degrees north this is pointed 30 degrees upwards, in a southern direction. Were the Earth spinning really fast falling would be more up/out than down.

Talking about “centrifugal force” usually invites arguments about whether it is a “fictitious force” we see because we use a rotating reference frame or whether this is needless pedantry.
xkcd.com/123/

One important aspect is that the acceleration is independent of the mass of a body: heavy & light things are equally accelerated. Sounds much like gravity but there it only happens because the inertial mass and gravitational mass are exactly the same. einstein-online.info/en/spotlight/i…

My physics teacher solemnly said this was a great mystery. However, Einstein resolved it some decades before by the equivalence principle: if gravity is the same as what you feel in an accelerated frame of reference, then they automatically become equal. en.wikipedia.org/wiki/Equivalen…

Falling in general relativity means following a geodesic (shortest path) through spacetime. My dropped marble curves along a path 30 m tall and 2.47 s long (about 734,000 km) in the curved spacetime near Earth. en.wikipedia.org/wiki/Schwarzsc…
numdam.org/article/AIHPA_…

It is somewhat non-intuitive and hard to visualize. I like Alessandro Roussel's brave attempt at visualizing it better. Not sure that it succeeds, but it is worth a try.

Another general relativity effect is that time flows a bit faster up here than down on the street. This is a factor of 3.33*10^-15. Over my lifetime so far this difference amounts to about 5 microseconds.