# Inertia Vs Momentum

At my recent lesson on kinematics, my teacher taught about inertia and momentum. This is what she said.

Inertia: a characteristic of an object that resists changes to its state of motion.

Momentum: The resistance of an object to a change in its state of motion.

My problem is what is the difference between both? I mean they look the same. And why does inertia not have a unit but momentum does?

Momentum: The resistance of an object to a change in its state of motion.

That sounds like a fishy definition of momentum to me. A slightly better definition, at least at your level, is that momentum represents the "amount of motion" an object has. Granted, "amount of motion" is a very vague term, but it stands to reason that if "amount of motion" were to be precisely defined, it would have the following characteristics:

• The larger an object, the larger its "amount of motion", all else being equal
• The faster an object is moving, the larger its "amount of motion", all else being equal
• If two objects have equal "amounts of motion" in opposite directions, the total "amount of motion" of the system of both of them is zero

Momentum is a precisely defined quantity that satisfies these principles. For slow-moving, massive objects, it can be calculated by $$p = mv$$.

A much better definition of momentum comes from the fact that it is conserved, but given the level at which you're currently studying, you probably haven't yet been taught enough background to really understand that, so I'll leave it out. (But do know that the fact that momentum is conserved in many situations is what makes it so useful.)

For inertia, on the other hand, this is quite reasonable:

Inertia: a characteristic of an object that resists changes to its state of motion.

The idea is that objects which are harder to move, or whose motion is harder to change, have more inertia. The precisely defined quantity that satisfies these properties is inertial mass, which you probably know as just "mass". (Or actually, energy, but at your level you can pretend it's just mass. You probably won't run into situations where this difference between mass and energy becomes important for quite a long time.)

Inertia is an intrinsic characteristic of the object related to its mass. Inertia tells you how much force it will take to cause a particular acceleration on the object.

Momentum is a function of an object's mass and velocity. Momentum is a measure of the kinetic energy of the object.

A massive object can have any momentum (at least as long as its velocity is less than light speed) including zero or negative momentum depending on the reference frame and coordinate conventions, but always has positive nonzero inertia.

• Nitpick on paragraph 2: you're not wrong about momentum being a measure of the kinetic energy of the object, in some sense, but that may be a little misleading because two objects can have the same kinetic energy but different magnitudes of momentum, or vice-versa. I'd say momentum and kinetic energy are two different measures of motion. Apr 28 '15 at 11:33
• If momentum is a vector, what is "negative momentum"? Just curious. Jul 28 '18 at 10:38
• @LarsH momentum in the negative direction of the chosen axis. Jul 28 '18 at 13:33

Good question. Clear thinking is essential in physics, and too many teachers and students just move forward to plug-and-chug equations without first spending enough time to study, explain, and fully understand the behaviors behind the math.

Inertia is a general concept that describes the observed behavior of objects in motion and at rest that Newton wrote out in his first law. It has no magnitude or measurement units. To quantify Inertia, physicists developed the more specific concept of Momentum, which has quantities of mass, speed, and direction.

Inertia is the observed natural tendency of an object in motion to keep moving in the same direction and at the same speed, or if at rest, to stay at rest. In physics, we quickly learn the the only difference between an object moving or being at rest is the relative motion of the observer, so both of these cases are really the same thing. Inertia simply tells us that to change the motion of an object requires applying a force to it.

To fully describe the property of Inertia with units we can measure, we must quantify an object's speed of motion, direction of motion, and resistance to change of motion. We combine these three things into a single parameter for each object called its Momentum.

The resistance to change of motion does not depend upon direction, so it is what we call a "scalar" quantity, and has been named Mass. We quantify Mass using units such as kg or lb-mass. We actually measure the Mass of an object indirectly by applying a known Force to it and measuring its change in motion (acceleration) according to the equation Mass = Force / Acceleration. Objects that require a lot of Force to accelerate a little bit have large Mass. Objects that accelerate a lot with a little bit of Force have small Mass.

Speed is also a scalar quantity, but when we combine speed with direction, we get a "vector" quantity called Velocity.

Momentum has both a magnitude and a direction. We quantify the Momentum of an object as the product of the Mass scalar quantity times its Velocity vector.

Momentum is also a measurable property of a sets of objects. Their individual Momenta can be added together using vector addition and be represented by a virtual object we call a Center of Mass moving with a Net Velocity.

Momentum is "conserved", which simply means that it does not change over time for any closed system unless some external force is applied. For a collection of objects, their collective Momentum does not change, even if they bang into each other and bounce apart again or clump together, or one object goes spinning off away from its neighbors. These collisions, if not perfectly elastic, will reduce the Kinetic Energy of the system, especially if they clump together into a single object, but the Momentum of the clump will be the same as the net momentum of all the original individual pieces -- the Center of Mass will continue to move with the same Net Velocity. By this example we can see that we should not confuse Momentum (Mass x Velocity) with Kinetic Energy (Mass x Velocity^2). While Total Energy is conserved (first law of thermodynamics), Kinetic Energy can shift into other forms such as Thermal Energy or Potential Energy, so it is not conserved like Momentum.

Note:

As an important caveat, the above applies to the realm of classic Newtonian physics where velocities are much, much less than the speed of light (speed of information) and relativistic effects are negligible. As objects approach the speed of light, velocity and mass are not so distinct, and additional energy appears to the observer to increase the mass of an object rather than its velocity. Still, momentum is conserved.

Inertia is what we simply called 'quantity of material'. The word material has been used here to specify the matter of body. For example, a plastic chair, a wood chair and an iron chair. Among them, a plastic chair will have less inertia because it will apply less reaction force, so it is easy to lift it. And the word quantitative is used to define the number. For example, a 10 kg wood and 10 kg iron would have same inertia.

Whereas momentum, $p$, is directly proportional to inertia, that means how much the body oppose the external that is what we called inertia, the more will be momentum. So we basically define inertia and momentum like this: the tendency of body to oppose the external force applied to change the shape size or position of body is inertia and the quantity of motion is called momentum. Mark the word external force.

There are good answers here already, but none of them seem to have addressed what you said about being told that momentum is a resistance of change of motion. That statement could be accurate, if you jump through a couple hoops in how you interpret it.

$$\frac{d\mathbf p}{dt} = \mathbf F$$

This is the definition of momentum, if you think about situations where the mass is effectively constant, like throwing and catching a ball, you get: $$m\frac{d\mathbf v}{dt} = \mathbf F$$ which you can again simplify and rewrite as: $$m\frac{\Delta v}{\Delta t} = F_{avg}.$$

So then it's pretty clear that $m\Delta v = \Delta p = F_{avg} \Delta t$, that is to say: the net change in momentum of object [the impulse] is equal to the average force multiplied by the time of contact. So if you imagine a ball with some momentum $p$, if the time of contact is constant regardless of the force required, then you can see that the force required to stop an object, to change it's motion, is directly proportional to it's momentum.

Also, note the $m$ in the expression for the impulse, you therefore have two things that can directly change how hard it is to change the motion of an object, the mass, and the product of the mass and velocity: momentum.

Inertia is an antiquated historical term, and it is not amusing to read tortured attempts to obscure that fact that its meaning is indistinguishable from the property of matter we now call inertial mass, which--in classical physics--is resistance to acceleration.

Mass is erroneously defined in many books and many classrooms, especially in chemical science, as "the amount of 'stuff,' or the amount of matter in an object.   Do yourself a favor and unlearn any notion you may have that mass is matter.   Mass is not matter; it is a property of matter.

Inertia is an outdated, unitless, useless, and confusing synonym for inertial mass.   I have thrown it out of my personal physics lexicon.   I'm a physics teacher and I don't teach inertia except as a historical artifact we have to endure where we find it still in use.   I just nod and smile I encounter it in print, or in conversations where argument about it is not warranted.

Momentum is the product of an object's mass and velocity, a quantity which Newton considered to be the fundamental quantity of motion.

From an outsiders perspective, I see inertia as the state of the matter, and momentum as measurement of that state. Inertia can be a state of non movement or extreme speed. The rate at which this movement or non movement is allowed to continue is momentum. That is why momentum can be measured but inertia can only be observed.

• Welcome to Physics SE :) we deal with mainstream physics here and as this seems to be rather uncanonical, sources for your explanations would be lovely. Jul 28 '16 at 21:56

basically one thing momentum is nothing its is change in acceleration nothing else guys use graphical interpretation to solve this problem u will feel calm when object starts moving it mean its acceleration changes it means it gain momentum that why it is moving if momentum is not conserved body will not able to move initially momentum begins to rise and inertia starts decrease obvious so momentum and inertia is inversely concept they are likely to be same but are very opposite . hope you enjoy with answer mechanics are change guys concentrate on real life problem which are happening in nature may be you can discover 4 lAw of motion .

• You need to better edit your answer: in its current form it is very very difficult to read and understand. Jun 24 '17 at 19:09

One of the first intriguing thoughts I had in physics is how can a small particle at high speed like a bullet do the big damage of a much bigger mass. What is the connection that makes speed a substitute for mass. It turned out that this is in fact, the most important question of the whole of physics. It is momentum that is at play here of course, which I consider to be the pillar of physics and the universe at the same time!! But how momentum that is intangible be related to mass which is very tangible so to speak. One of the indirect replies to this is Einstein equation E=mc^2. But this is energy and mass with no momentum. Well, mass and momentum are related- since the integral along a line of p.dp is p^2/2 and we know that E=.=p^2/2m. Thus energy is nothing more than a line integral of momentum and energy is equivalent to mass as per Einstein, and of course mass is related to inertia. So mass, energy, momentum and inertia are well connected by well established formulae. Energy is conserved because it is an integral of another conserved quantity- momentum.

The next question is why momentum is conserved? Well momentum is conserved because the universe has a symmetry according to Noether theorms and defends this symmetry with the utmost vigour. For a simpler statement; This is because it is impossible to move the centre of mass of an isolated system from within. That is equivalent to saying that for an isolated system; sum(m.dx)=0 along any line in space. So if you move one mass to one side you need to move an equivalent to the opposite side to keep the world from tipping over!! And this is true even for something as light as an electron or a photon. Next, differentiate the sum with respect to time for constant mass and you get; sum(mv)=sum(p)=0, along any line ins space. Another differentiation gives; sum(ma)=sum(f)=0 along any line in space. Thus we see that the conservation of momentum p, and the action and reaction equality of forces are all of the same origin- the symmetry of space.

Now we see that if we wanted to move an electron without moving another, we have to move the rest of the universe a proportional distance to the opposite direction!! and this is inertia.. or what Mach's called it; the effect of distant masses. This becomes more accentuated when we do rotation, then the whole universe must rotate the opposite direction(reduced by its mass of course).

But mass is different than energy or momentum as it tends to exclude the presence of any other mass from the position it is sitting at. While energy doesn't have this property. Also mass can have other attributes like; spin, magnetic dipole moment, charge, etc that motion doesn't have. The answer to this is; because mass comes from rotation and all such attributes follow.

The simplest way to understand all the complications that follow, is to think of radiation as evaporated matter and matter becomes just condensed radiation that does the condensing by going round in closed loops. Radiation as it is carries many of the attributes of matter as any vapour should. It has electrical attributes in the form of electric and magnetic fields, but it also has mechanical attributes in the form of momentum and energy. Moreover, it obeys conservation of momentum like a mass.. that is it has inertia too. So that if you fire a gun of laser, you can hit and move any small mass at a distance, and also causes a recoil of the gun.

But can radiation go in closed circles. Apparently yes and in the form of a topological electromagnetic soliton. The soliton and anti soliton created by radiation can be a model of the process of pair creation from radiation. So to close, inertia is an expression of momentum conservation which is a specific to matter in its both condensed and evaporated phases. And all that is due to the symmetry of the space we live in.
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