If I attach the same engine to a 1,000 kg spacecraft and a 10,000 kg spacecraft in orbit, the 1,000 kg spacecraft will accelerate more quickly. If I drop a 1 kg rock and a 10 kg rock on the moon, they accelerate at the same rate. What is the difference?
I think what I may be asking is “why is gravity the a and not the f in f=ma.”
EDIT: BY all means please feel free to discuss, but I consider the question answered by u/mmmmmmBacon12345
mmmmmmm….. Bacon…..
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I’m no expert but I believe the official answer is “we don’t know.” Ask the top theoretical physicists and they can’t explain gravity other than “it just works the way we have observed.” Redditors, please correct me if I’m wrong.
Because the force of gravity is proportional to the mass of an object. Gravity *is* the F in F = ma, but it just so happens that an *m* also shows up in that F in such a way that the two m’s cancel out and you always get the same a.
…or at least that’s the classical Newtonian-gravity picture. Relativity changes the story somewhat, but the notion of acceleration in relativity is pretty different from how it is in classical mechanics.
Because the gravity-force depends on the mass of both objects: F = M1*M2*G/R^2 where M1 and M2 are the two objects’ masses, G is the gravitational constant, and R is the distance between the centers of the two masses M1 and M2. So when you put that together with the acceleration equation, you get this (assuming Earth is M1 and the object is M2)
F = M2*A = M1*M2*G/R^2
A = M1*G/R^2
The mass of the Earth (M1) and the gravitational constant (G) are relatively constant, so the only real factor for the acceleration of gravity is how far from the Earth you are.
On the other hand, the thrust of the rocket engine has nothing to do with the mass of the rocket. It pushes with 1000 Newtons whether the rocket has a mass of 1 kilogram or 10^6 kilograms.
F = M*A
1000N = (1kg)*A
A = 1000N/1kg = 1000 m/s^2
F = M*A
1000N = (1000000kg)*A
A = 1000N/1000000kg = 0.001 m/s^2
The gravitational force on an object is proportional to that object’s mass, but so is the object’s inertia. If you increase an object’s mass tenfold, you increase the gravitational force on it tenfold—but you also make it ten times harder to accelerate, and these factors offset one another perfectly.
You’ve said it yourself by quoting “f=ma”: the force (f) grows by a factor of ten, but so does mass (m). Since they’re on opposite sides of the equation, they cancel each other out. Better yet, rewrite it as “a=f/m”: increase the mass and thereby the gravitational force, and your acceleration increases by a factor of 10/10, which simplifies to 1/1, which is just 1, which means your acceleration remains the same.
The difference is that a mass at rest wants to stay at rest. The object that you are thrusting has to fight it’s rest state. The object that is falling into a gravity well is at rest the whole time while it’s canvas is being guided towards it’s source of attraction.
The force of gravity is F=G (m1 M2)/r^(2). The mass of the rock and the mass of the Earth both matter for the force
This means that the 10 kg rock experiences a 10x greater force than the 1 kg rock but since it’s mass and inertia are also 10x greater they both experience the same acceleration
When you slap a rocket engine on the two different space ships it’s the same engine so now the force is the same. Since the force is the same but the masses are different then the acceleration must be different
If you instead scale it up and slap 10 engines on the bigger one then the bigger one will again have 10x the force and the acceleration will match
F is the F in f=ma
The force of gravity is Fg=G*M*m / r^(2)
So A = F/m as always.
A = Fg=G*M*~~m~~ / r^(2) / ~~m~~
leaving
A = Fg=G*M / r^(2)
The only mass that matters is the other object, not the accelerating object.
This because mass does double duty. It has an attractive force towards another object, but it ALSO resists acceleration.
It just so happens that the gravitational mass (in Fg) and inertial mass (f=ma) are the same. And that is puzzling, and we don’t know why. There is currently no reason that *has* to be true, it just is empircally true.
So, a brief take on your scenario: Your engine produces a fixed amount of thrust, this is why a larger rocket accelerates slower.
Gravitational attraction is not fixed, it increases when you have more mass. So a larger rocket feels more gravitational force (2x the mass, 2x more force, 2x more acceleration). But larger rockets are *also* harder to move (2x the mass, 2x less acceleration), so in the end you get the exact same result for every object attracted to the mass in question (a.k.a. the planet)
There are two ways to look at this. The first is the classical way, Newton’s laws of motion and gravity.
The force due to gravity is “coincidentally” proportional to the mass. Compare this to electrostatic forces. The electrostatic force is proportional to electric charge, so that, in a given electrostatic field, an object with a strong charge and a small mass will accelerate a lot, while an object with a weak charge and a large mass will accelerate only a little.
With gravity, the “gravitational charge” is exactly proportional to the mass. So if you double the mass, you double the “gravitational charge”, and therefore double the force. When you double the force and double the mass, the acceleration is the same.
In classical mechanics, the “gravitational charge” being exactly proportional to the mass for any substance is a “coincidence”. There is no explanation for it, but for centuries all experiments confirmed it.
The second way to look at it is with general relativity – Einstein’s great theory. Here, gravity is not a force like electromagnetism (or chemical stuff, which is really just electromagnetism). Instead, objects in space bend space and time itself, and then other objects move along that bent space in what seem to them to be “straight lines” (straight in an obscure sense).
It’s a bit like how planes follow great circle routes when traveling long distances. Great circle routes are sort of straight lines on a globe, but they look like curves on a flat map. Objects move through this warped space-time in what look like four dimensionally straight lines to them, but looks like acceleration to us.
This is how general relativity provided an explanation for why gravitational acceleration does not depend on mass. The objects are just following these odd straight-ish lines.
As others have pointed out it takes more energy to accelerate objects that have more mass.
However it should be noted that gravity does scale the with the mass of an object dropped onto another object. It is the combination of the mass of both objects one needs to consider. But if one object is the size of the earth or moon and the other is only a few tons, the mass of the smaller object can be ignored.
Look at it this way. What would you have to do to the engine to make the 10,000 kg and 1,000 kg orbital masses accellerate at the same rate in space, say at 9.8 m/s^2 ? You would reduce the thrust for the smaller mass, or increase thrust for the larger mass, so that each object experiences exactly the forces needed (the same forces that Earth’s gravity would exert on each of them) to achieve 9.8 m/s^2 acceleration.
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