The More Massive They Are the More Forceful They Fall

There is a common (mis?)conception, if not law, that declares things fall at the same speed regardless of their weight. For instance, Galileo Galilei dropped a cannon ball and a shot off the Tower of Pisa and each hit the ground at the same time, disproving Aristotle’s theory that things fall with proportional speed to their mass. Aristotle was further disproven and Galileo proven, when the astronaut David Scott dropped a feather and a hammer in the near vacuum of the moon and the two objects landed at the same time. Here on Earth, that effect is only altered by air resistance as the shape of the object will determine it’s aerodynamics and terminal velocity.

This is all well and good but in direct conflict with Newton’s inverse square law of gravitation which follows Aristotle’s logic that objects will attract with a force proportional to their mass but also inversely proportional to their distance squared. The main point to be noted here is that with more mass comes more attractive force; with more attractive force comes more acceleration; with more acceleration comes more velocity; with more velocity comes less time falling. If you believe Newton you cannot believe Galileo and could believe the “disproved” Aristotle.

I posit to believe Newton and some of Aristotle over Galileo and Commander David Scott. Albeit, I don’t believe Aristotle fully as I’ll come back to later, but just partially for his logic that objects will fall proportionally to their mass. And it’s notable Galileo and Commander David Scott are not outright wrong or deceitful either. So, perhaps it’s easier to begin with why Newton disproves Galileo and Scott to show who is right here.

For reasons I’ll avoid for succinctness, Newton has been proven right over and again that his theory of Gravity is correct on a macroscale. From orbital observations and modelling to the impetus for Einstein’s fine-tuning of gravitational theory we can presume that Newton’s formula of one mass times another mass over their distance squared will give us the force of the two bodies’ gravity for one another. With that, it is unavoidable that two objects with greater total mass will supply a greater force and thus speed of attraction. What is going on here with Galileo’s and David Scott’s solid proofs of falling objects? They do not account for the minute discrepancy that is imperceivable to the observer. Human or otherwise, when an object as massive as the Earth or the Moon are interacting gravitationally with an object as massive as a tiny cannon ball, feather, or hammer the change in force is negligible. What that means is that the mass of a feather and the mass of a cannon ball are effectively equal when measuring their gravitational force toward a planet and so fall with the same acceleration. However, technically the mass and thus the gravitational acceleration does differ.

So, Aristotle came to his idea based on intuition and not with knowledge of mass, force, and gravitational interactions. He did not regard the negligible difference that exists on a microscale but rather thought the difference existed on a macroscale. What he was thinking of was more along the lines of air resistance and aerodynamics, and was outright ill-conceived. What Galileo conceived of was also intuitive, but he was right for the wrong reasons in a way. Galileo figured the attractive force and thus fall-time was equal in absence of air based on observations and experiments. He was correct if you negate the minor discrepancy that is near impossible to perceive. He is wrong if you are an anal bastard for physics, measurements, and mathematics.

So, you can rest easy knowing things fall the same regardless of their mass if you are concerning yourself with Earthly objects. It will not affect you to know that technically, when you throw a ball in the air the ball is attracted to the Earth but the Earth is also attracted to the ball. The attraction to the ball is negligible due to it’s tiny scale and so the Earth is not effectively displaced by the upward thrown ball. It will not affect you when a feather and a hammer fall and the hammer is falling with a bit more acceleration, air resistance takes over the majority of the event. Even in a vacuum, if you drop two things of varied weight, they’ll land at the same time to within an irrelevant amount of time. However, if you are calculating the gravitational force and velocities of planets in orbit or black holes merging you will see this tiny discrepancy disappear as you increase the scale beyond the mundane workings of Earthly gravity. Fortunately, you can keep your acceleration due to gravity on Earth as 9.8 m/s² as it accounts for the mass of the Earth which is approximately 5.972 × 10²⁴ kg and thus renders your 22 oz hammer at 0.6236 kg negligible.

In the end, I think everyone in this story is right one way or the other: Things fall faster proportional to their mass but the difference is negligible for folks’ watching little things fall down to Earth… or the Moon.

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