# Why does mass limit acceleration?

If a force of $$10\,\mathrm{N}$$ is applied to different objects of different mass in empty space, in the absence of gravity, why do lighter objects accelerate faster than heavier objects? Why does mass cause inertia?

• Related, from earlier this week: Why did we expect gravitational mass and inertial mass to be different? Apr 19 at 21:04
• If you have any sort of fundamental answer for this, stand by for a phone call from Sweden.
– J...
Apr 20 at 11:36
• @J... Alas, the fundamental answer is called Noether's Theorem, and was discovered quite some time ago. And no, Emmy Noether didn't win the Nobel Prize for this either. Apr 22 at 0:27
• @MattThompson how does the Noether's theorem explain this? Apr 29 at 14:53
• @MattThompson Noether's theorem assumes Lagrangian description which already encodes Newton's 2nd law in it. You're not providing a fundamental answer for the origins of inertia, just rewriting it in a different mathematical formalism Aug 18 at 1:24

I do not think it is mass that "CAUSES" inertia. Rather mass is DEFINED to be the property of an object which gives it inertia

For example, imagine that you were a theoretical observer who did not know of the concept of mass. There were just 3 objects in front of you. You applied $$10 \mathrm{N}$$ to each object and observed that the different objects accelerated at a different rates. One accelerated at $$1 \mathrm{m}/\mathrm{s}^2$$. Others at $$2\mathrm{m}/\mathrm{s}^2$$ and $$4\mathrm{m}/\mathrm{s}^2$$ respectively. So, you can conclude that the different objects have some property that determines how much they accelerate under a given force. You would also notice that there was a linear relationship between the force on a body and its acceleration.

Now, you can calculate what would be the value of this property. And you can see that the body that accelerated at $$1 \mathrm{m}/\mathrm{s}^2$$ would have a higher value of this property than the one that accelerated at $$4 \mathrm{m}/\mathrm{s}^2$$.

The theoretical observer can see that this property is directly related to what people call "heavier" and "lighter" objects. So, it is safe to conclude that lighter objects accelerate faster than heavier objects.

To summarise, the reason lighter objects accelerate faster than heavier objects is because that is how we have defined the terms "lighter" and "heavier".

Mass "causes" inertia, because that is how we have defined "mass".

If you want to further ask, why objects with different values of this "mass" property accelerate differently, then you are asking a more fundamental 'why' question. And like most fundamental why questions in physics, it boils down to , because those are the laws of physics in our universe. And if you want to ask why are the laws of physics like that, the answer is " because of the initial conditions at the Big Bang. And researchers are working to figure this out "

EDIT : As some comments have pointed out, there is also the quantum mechanics aspect of this.

All particles get their inertia or the "tendency to resist motion under a force", through the way the fundamental particles interact with the Higgs field.

To understand why massive objects have inertia, we need to understand where the mass of objects comes from. We know that all massive objects are made up of atoms. So, atoms give objects their mass. But where do the atoms get their mass from?
Atoms are made up of protons and neutrons (and electrons but they have negligible mass). So, protons and neutrons give atoms their mass. But where do protons and neutrons get there mass from ?

According to the standard model, protons and neutrons are made of quarks. So, quarks should be giving atoms their mass, right ? WRONG.

Quarks have substantially smaller and lighter mass than the protons and neutrons that they comprise. It is estimated that the masses of the quarks, derived through their interaction with the Higgs field, account for only about 1% of the mass of a proton, for example.

So, 99% of the mass of a proton is not to be found in its constituent quarks. Rather, it resides in the massless gluons that bind together the quarks inside the proton. These gluons are the carriers of the strong nuclear force, that pass between the quarks and bind them together inside the proton. These gluons are "massless" but their interaction with the Higgs field is what we experience as mass.

So, at a quantum level , the Higgs field ‘drags’ on the gluons , as though the particle were moving through molasses, (Another analogy used is moving through a crowded dance floor) . In other words, the energy of this interaction is manifested as a resistance to acceleration. These interactions slow the particles down, giving rise to inertia which we interpret as mass.

• @my2cts I don't think you need to invoke the Higgs Mechanism to answer the OP's question. This answer is an entirely satisfactory classical explanation of mass and inertia. Apr 19 at 16:11
• And why is this concept that we are now measuring exactly the same (ratios) as the gravitational force between the objects? Strange indeed. Mass tells space where to be and space tells mass where to go. Apr 19 at 16:11
• the initial conditions at the Big Bang are unrelated, they are consequences of the physical laws not the cause if anything Apr 19 at 17:02
• That an object which has more gravitational mass also has more inertial mass is not something that is defined; it's something observed in the world.
– Kaz
Apr 20 at 2:11
• I would want to more carefully word this answer, because right this second you have both statements: "I do not think it is mass that CAUSES inertia... Mass causes inertia, because that is how we have defined 'mass'." Apr 20 at 5:06

In classical non-relativistic physics, the inertial mass $$m_{\mathrm{inertial}}$$ is by DEFINITION the quantity of an object that acts as the proportionality factor relating force $$\vec{F}$$ and acceleration $$\vec{a}$$ in $$\vec{F}=m_{\mathrm{inertial}}\vec{a}.$$ We observe that different objects react differently to the same force, i.e. having a different acceleration, therefore we assign different masses to them.

There is also the gravitational mass $$m_{\mathrm{grav}}$$, that gives a notion on how much an object is affected by gravity, therefore kind of acting as a charge of gravitation. All experiments show that gravitational and inertial mass are the same, hinting $$m_{\mathrm{grav}}=m_{\mathrm{inertial}}$$. This is hinted by the observation that all objects have the same acceleration in a gravitational field, implying that the masses in the equation $$\vec{g}m_{\mathrm{grav}}=\vec{a}m_{\mathrm{inertial}}$$ cancel; where $$\vec{g}m_{\mathrm{grav}}$$ is (approximately) the gravitational force asserted on a body with gravitational mass $$m_{\mathrm{grav}}$$ at sea level on earth. Why this should be the case, theoretically, was not explained in non-relativistic physics, but nowadays by the equivalence principle in general relativity.

• It is crucial to explain as you do that the real question is why inertial mass and gravitational mass are the same. The answer is "because of equivalence principle". That might cause someone to ask "How does equivalence principle explain why inertial mass and gravitational mass are the same?" Apr 20 at 20:03
• @PanuLogic lol good point. Aug 17 at 23:09

We call mass, inertial mass because it is the efficient (main) cause of inertial motion. How could there be motion if there was no time or no space? Hence, inertial mass is not the only cause, space is required as well as time.

Aristotle called this natural motion, it's the motion natural to an object in virtue of simply being itself.

We find out by experiment that a force of $$F$$ (measured in Newtons) on an object of mass $$m$$ (measured in kg) causes an acceleration of $$F/m$$ (measured in $$m/s^2$$).

Hence, lighter objects accelerate more quickly than heavy ones. It's an outcome of experiment. This is because physics is an experimental science and an empirical inquiry into the nature of space, time and matter.

• How can mass be the cause of motion? Also in special relativity mass is defined as rest energy divided by $c^2$. This overrules Newton's definition. And perhaps General Relativity should also be mentioned. Apr 19 at 15:23
• @mycts: Without mass, space & time there can be no motion. Look up Aristotles notion of efficient and material cause. The revisions you point out make no difference to the notion that mass is the cause of motion. Einstein himself said that he viewed the geodesic law of motion on a spacetime manifold not as so many commentators tell it, that is as an aspect of geometry, but as a unification of inertial motion and gravity. Apr 19 at 17:23
• Inertial motion is the same for all bodies (constant vector of velocity in special relativity, more generally motion along geodesics), hence mass plays no role. Inertial motion is purely an artefact of the choice of coordinate frame. It is only when forces are applied, i.e. when the motion is non-inertial, that inertial mass plays a role. So your logic seems to fall short. Apr 20 at 7:13
• @tobi_s: If there is no mass there can be no motion, there is nothing to move. You say "inertial motion is the same for all bodies" - are we to suppose that yhese bodies have no mass? Can you show me such 'bodies'? Apr 20 at 8:42
• @MoziburUllah You sure seem to be playing word games here. "Inertial motion is independent of mass" is a scientific fact. You said "mass is the main cause of inertial motion" which is a philosophical statement. So we find that inertial motion is independent of what you consider its main cause. That was my point. I also deliberately said "bodies" to avoid the sidetrack discussion that you are trying to start now. Apr 21 at 23:46

Maybe it is because an object with more mass has to bend spacetime more in order to move through it.

• I don't know if it is true, but your answer would make great intuitive sense Apr 20 at 20:05
• Oh yeah really this makes sense Apr 23 at 3:25
• That explains the extra gravity on massive objects, but not the resistance to acceleration. Aug 17 at 21:29

More mass means more stuff. More stuff is harder to push than less stuff, so you need to push harder to get the same acceleration.

The reason for this can be understood in terms of momentum. Imagine for a moment that mass does not limit acceleration, and that we instead have the physical law:

F = a

If I push a book across a frictionless desk with a force of 1 Newton, it will have a velocity of 1 m/s after one second. If I push a stack of 10 books across my desk with the same force, it would have the same speed of 1 m/2. If I push a freight train loaded with 100 trucks of books across a frictionless track, it too will have a velocity of 1 m/s with only 1 Newton of force applied for 1 second. In reality, we would expect the freight train to be much harder to push, no?

This is where momentum comes into play. In some sense, momentum is the amount of "oomph" an object has behind it. It is also a physically conserved vector quantity, which means the amount of momentum in the universe never changes. Momentum, p, is given by:

p = mv

Where v is velocity. We can express force in terms of momentum by:

F = dp/dt

That is, the derivative of momentum with respect to time. Note here that in the second equation there is no explicit relationship with mass.

Momentum is what we call an extensive quantity, which means the more we have of something, the more momentum we can expect. If two cars travelling in opposite directions collide head-on their opposing momenta cancel and they will end at rest, close to the impact site. On the other hand, if a car hits a freight train travelling at the same speed in the opposite direction head-on, the speed of the freight train will barely change (until the operator applies the brakes). Its momentum will be reduced by the change in momentum of the car colliding with it, which will be a small fraction of its initial momentum. In this sense, the mass is simply the amount of the thing carrying the momentum, so more "thing" means more momentum (for a given velocity).

The long answer Part II - Energy

This can also be explained if we think in terms of energy.

For a constant force on an object, the energy is the force times the distance the force was applied over:

E = F*d

No matter how massive the object is, the energy will be the same. If we split the large object into two equal parts after accelerating it, each will have the same speed, but half the energy.

Now, the energy from motion (kinetic energy) is given by:

KE = 1/2 mv^2

So, if we apply a force F over a distance d for two objects with different masses, we must ultimately get a different final velocity in order for them to have the same energy. In this case, the smaller object must be accelerated more quickly to achieve the higher final velocity.

For the final velocity to be independent of mass we would need to decouple force from energy.

Energy is another quantity that is conserved in physics. I won't go into it, but the many relationships that exist between velocity, momentum, energy and force can ultimately be derived from the conservation of energy and momentum. These conservation laws can further be derived from the assumption that the laws of the universe are constant over all time (energy conservation) and space (momentum conservation). This proof is called Noether's Theorem, although it may be a little difficult to grasp for a beginner.

• This answer deserves more up votes. Elegantly simple, and also correct. Apr 21 at 11:07
• I added a discussion about momentum which may be more intuitive, as well as some commentary on "extensive properties" in physics. F = dp/dt was actually Newton's original expression for his physical law, but it is often introduced to physics learners as F = ma as it includes neither calculus nor momentum (which is a concept introduced later). The idea that F = ma is the definition of mass is simply wrong. It follows from momentum and energy being extensive properties, which is a logical necessity for the laws of physics to make any sense at all. Apr 22 at 0:22
• In this case, "mass" simply means "how much". How much "of what" isn't important, at least not for the classical description. Apr 22 at 0:24
• Kinetic energy relates more to F=ma definition. The kinetic energy equation is directly from $F=ma$ as $E= \int F dx = \int m\tfrac{d^2x}{dt^2}dx=\tfrac{1}{2}mv^2$ Aug 17 at 23:07

I am only going to talk about massive particles as in your example.

There are two ingredients:

1. Higgs mechanism

Today we do know that we live in a universe that is permeated by fields all through. This includes the Higgs field.

Now the Higgs mechanism is a true phenomenon in that it shows you that when certain particles (which themselves are excitation of their own fields) couple to the Higgs field, we can notice something special in experiments. This is that they (certain particles) acquire rest mass.

Now in your example, you apply 10N of force to let's say a small and a huge rock. The small has less number of particles (excitation of those fields) that couple to the Higgs field.

Now imagine both rocks being made up of quarks and electrons (simplified), and to be very intuitive, lets imagine these electrons and quarks as little ships with anchors.

The bigger rock (assuming same density) has more anchors, and feels a bigger resistance to your force of 10N. This resistance to acceleration is what we call rest mass and is a beautiful tethering effect.

1. Energy

Now please note that in the case of complex objects, the rest mass is also (mostly) due to binding energies. This is astonishing, but in the case of a nucleon (proton or neutron), the rest mass is 99% due to binding energies. Thus, the true answer to your question is stress-energy (just like in general relativity).

An object, like a rock in your example possesses stress-energy (mostly in the form of binding energies), and this is the true source of its rest mass, which we see in experiments as a resistance to acceleration. In this sense, the answer to your question is that rest mass, the resistance to acceleration is rooted in the energy that the object possesses. The larger rock in the example possesses more energy, more resistance to acceleration. The more fundamental question is why more energy causes more resistance to acceleration, and the true answer is nobody knows, but this is what we see in experiments.

• I do not really understand, why the Brout/Englert/Higgs (BEH) mechanism is brought into this. The rest mass of the electron is 0.5 MeV/c^2, the one of the quarks inside a proton around 10 MeV/c^2. The proton weights around 1000 GeV/c^2, therefore the BEH mechanism accounts for only around 1% of the rest mass we experience in everyday life. The rest, around 99% of the mass, is binding energy. Therefore I do not think any analogy to visualise the BEH mechanism answers the question. Apr 19 at 22:36
• @Koschi By your numbers that would make it 99.997% Apr 22 at 11:42
• @user253751, Sorry, there is a typo in my comment. The proton of course weights around 1 GeV, not 1000 GeV. Apr 22 at 13:04

The answer is not higgs,It only accounts for only about 1% the mass of the proton. The real answer is energy.To know how energy causes resistance to motion,see this video:https://youtu.be/gSKzgpt4HBU Here,he explains how a system of photons (which has no mass) aquire mass by constraining its energy of motion.This is what we mean by $$E=\gamma mc^2$$

From Newton's second law of Motion we can say,$$\vec F = m \vec a$$

Upon rearranging the terms we get ,

$$\vec a = \frac {\vec F}{m} ..... 1$$

We know that acceleration is the rate of change of velocity , but from above equation we get another definition of acceleration i.e

Acceleration is the Force acting per unit mass.

Now when you apply the same force on a heavier and a lighter body the value of force per unit mass is more for the lighter than for heavier body so lighter body has more acceleration that heavier body.To look at it more intuitively ,

The force applied by you is dstributed over less mass in case of lighter body so each unit mass is acted upon by more force and hence more acceleration. So acceleration of each particle is more.However the same force gets distributed over a large number of unit masses in the heavier body so each unit mass is acted upon by less force and hence less acceleration.

Mass is more than just resistance to motion. It is a measure of three factors that agree, with no immediate and obvious reason they should:

1. Inertial resistance to motion under force

2. A property of mutual attraction that makes two objects with mass attract - inversely proportional to square of the distance (just like with electric charge, magnetic dipole, and the strong nuclear force).

3. The quantum wave’s debroglie rest mass, of matter-energy.

Several have said mass is nothing more than an extensive property we assign to objects based on. Truest form to Newton’s second law is not actually $$m:= \frac{F}{a}$$

Definition 1 of mass: $$\mathbf{m:= \frac{\int F dx}{\Delta v}}$$

But this missed the other aspects of mass.

Gravitational mass always equals inertial mass. There is no immediate, obvious reason this would need to be true. Until physicists made advances in relativity and particle physics, there was no known reason. In fact that is another definition of mass:

Definition 2 of mass: $$\mathbf{m:= \tfrac{Fr^2}{Gm_0}}$$, plus assignment of a definitional benchmark mass $$\mathbf{m_0}$$ to a neutron at rest.

This is similar to the definition of electric charge and strong nuclear force parameters. For example, compare to:

Definition of charge: $$\mathbf{q:= \tfrac{Fr^2}{k_eq_0}}$$ , plus assignment of a definitional benchmark charge $$q_0$$ to an electron.

It is this definition of mass is the most normal and consistent with other fundamental forces in classical physics. What’s unique about mass is that, unlike with other forces such as electric charge, this same parameter controls other properties, which would be surprising if we weren’t used to it. (And it would still be considered given, or a fluke, if not for general relativity and quantum mechanics ultimately reconciling them.)

Mass always implies energy-matter, and not simply in the way that all fundamental forces create energy. A particle with mass accelerates another particle giving it energy, and the energy is $$\tfrac{1}{2}mv^2$$. While a charged particle does accelerate another particle giving it energy, the energy is not expressed as $$\tfrac{1}{2}qv^2$$. This is discussed to distinguish kinetic energy from energy-matter to clarify that it’s a third, separate aspect. Kinetic energy relates more to the above definitions. The kinetic energy equation is directly from $$F=ma$$ integrating $$E= \int F dx$$

More fundamental than K.E., the mass is a store of energy itself, not just a carrier of energy while in motion due to its inertia.

Definition 3 of mass: $$\mathbf{m:= \tfrac{E}{c^2}}$$

## Summary

Any of these definitions fully defines mass, and there are no immediate reasons that

$$m_{inertial}=m_{gravitational}=m_{energetic}$$

should necessarily be true. A good understanding of general relativity and quantum mechanics is necessary to know why.

• The special theory of relativity may be used to derive that $m_{inertial} = m_{energetic}$ (hence Einstein's famous $E=mc^2$). The general theory of relativity explains why inertial and gravitational mass are the same. Aug 18 at 0:41
• @EricSmith “There is no immediate reason this would need to he true. Until physicists made advances in particle physics, there was no reason.” was my attempt to say it is ultimately reconcilable, just not immediately and obviously. And not before 1900. I will change ir to “advances in relativity” Aug 18 at 1:13
• But wasnt very clear. I made that change. I also added the final sentence, and a third pointer to that fact. As far as I saw, my answer is the only one that identified these three aspects of mass at all. Only one even referenced gravity. Another said kinetic energy was a completely independent aspect (but just comes from F=ma) Aug 18 at 1:22
• What about adding $$m := \frac{ \int F {\rm d}t}{\Delta v}$$ as a definition, describing the relationship between impulse and speed change. Aug 18 at 11:15
• @JohnAlexiou i think that would be another version of F=ma ?? Not certain Aug 19 at 22:37

It doesn't, or not really and certainly not in your example.

In a closed inertial system forces may apply themselves, but then their effect can only be determined by also translating whatever it is the force is applied on into pure force only, since on the applying side there is nothing other than force. Its not a real world situation, so it doesn't really matter which 'side' applies the force and which one accelerates. Much mass represent much force. Small mass represents little force. Hence the difference in the resulting acceleration. However, the idea the larger mass results in less acceleration than the smaller one is not always true. This may sound a bit confusing, but in a closed inertial system, your 10 N itself accelerates considerably faster when applied on a large mass, then when applied on a small mass.

It appears the force needed to push something is equal to the force needed to pull it, which is a mere consistent observation. In an expanding universe, this should not be so, but apparently it is. Why it is so, nobody knows. It indicates expansion due to difference is not omni-directional and contraction due to similarity does not result in a single vector. Translated into every day practice terms, that would mean the options to get away from something exactly equal the options to get towards something, which does not make sense. Nevertheless, that's how it is. This indicates that the reason infinity is really just a product of imagination, is not because nobody has ever been there, but because it doesn't exist as such. Apparently far away has a back door, which in itself is encouraging.