# What is actually meant by momentum at the deepest level of physics?

What does something called momentum actually measure? I know that generally momentum of a object is describe by the multiplication of mass and object's velocity and it is a conserve quantity without any presence of External force but what does really mean by External or Internal applied force on a object in the angle of momentum?

• I think a problem with classical thinking is that position and velocity are the "important" things. Actually it is relative position and relative momentum--not really velocity. We tend to "see" momentum as a velocity--what's really important is not the velocity but rather the momentum. Internal transfer of momentum describes such things as molecular particles bouncing off of each other and transferring momentum, internally, which results in no net increase/decrease of momentum. External momentum is something that transfers momentum from a large system to another. – Jared May 10 '16 at 3:07
• Linear momentum is the Noetherian current associated with the invariance of physical law under translations in space. But you aren't yet ready for that description, so you have to live with "momentum tells you the product of applied force and time of application needed to bring the object to a stop in the same frame where the momentum is calculated". – dmckee May 10 '16 at 3:25
• I have to agree with @dmckee, the non-trivial side of momentum is its conservation due to the homogeneity of space, but before you can understand that, you also need to develop a working intuition for it that is based on its phenomenology. Just like you need energy to perform work (i.e. apply a force for a certain distance), you need momentum if you want to get a massive body moving. The more mass the body has, the more momentum needs to be available. The higher a velocity we want to achieve, the more momentum we have to exchange. This may sound trivial, but that is what momentum quantifies. – CuriousOne May 10 '16 at 4:04
• @CuriousOne : these comments (by Jared, dmckee and yourself) seem to be answers rather than requests for clarification etc as in the Help Centre ("When should I comment? When shouldn't I comment?"). If you are answering the question you should post an Answer. – sammy gerbil May 10 '16 at 19:32
• Downvoted for poor title. – Tomáš Zato May 11 '16 at 12:40

## 2 Answers

You are studying the very basics of mechanics. In that case, as @dmckee♦ quoted, we will discuss it on the fundamental level.

Momentum is the characteristic property of a moving body. There are many dynamical variables associated with the motion of an object, like the displacement, velocity, acceleration etc. But, these quantities are just only variables of the motion. They cannot tell you the aftermath of the motion. For example, you heard someone telling a man on the pavement is hit by a car. You cannot get into a clear image unless you know the velocity of the car. Now, if you hear that the man is hit by an object at a velocity of $5 m/s$, still you didn't get a clear idea about the situation. You need to know how big that object is. If the object is a small mass like a stone, it will not hurt that much. But if it is like the size of a truck, then the injury will be maximum. So mass and velocity has to combine to give rise to a new quantity that could explain the impact a moving object could make on another. This way, we could visualize, how the energy or force is being transferred from one body to another. This new quantity which is the characteristic of a moving body is what we call the momentum of the body.

Momentum is defined as
$$\vec{p}=m\vec{v}$$

So momentum is of great importance in collision theories and all places where all we need to study about energy transfer. As you can see, the momentum is not a fundamental quantity like the displacement. It's a derived quantity. The velocity is a vector quantity and hence a scalar (mass) times a vector (velocity) gives you a new vector (momentum) in the same direction as the original. So the momentum (we are talking bout the linear momentum) is a vector quantity whose direction is determined by the velocity of the moving body. The peculiarity of a vector quantity is that it requires both direction and magnitude to represent the quantity completely. Unlike the solo quantities like displacement, velocity etc, the momentum (which is mass times velocity) tells you the effect of motion.

You can see that, when the velocity of a massive body is zero (i.e., it is at rest), then the momentum is also zero. It doesn't matter whether it's mass is huge or not; since it is not moving, it's momentum is zero. That's why we say momentum is the characteristic property of motion. Also, since the object is at rest, it will have zero kinetic energy an so the object cannot transfer it's energy to another one. So, the momentum tells you the direction along which energy transfer takes place. Let's see an example.

Suppose you are a football player. You have a ball and you just kick and it will move. So, the ball gains kinetic energy as it's velocity increased. But, from where, the kinetic energy came from? It's from the kinetic energy of your leg. When the leg collides with the ball, at the same instant, the leg imparts a momentum on the ball and by that way, the object transferred it's energy. So energy is transferred by transferring momentum. Your leg is moving with a certain velocity. So it has a well defined momentum. If it hit the ball, then the momentum is transferred to the ball. It's initial velocity was zero. So the ball gains momentum and it has now got a velocity and since it has got a velocity, we say the kinetic energy of the ball increased from zero. So energy is transferred from the leg to the ball. Also, to observe the vector property of momentum, you can see that the momentum of the ball is in the same direction as the velocity of your leg. That is, the ball fly off in the same direction as you kicked.

Now, what if the object having a less mass and less velocity hit on a wall? For example, a person running at $5 m/s$ hit on a wall. The wall will not move, of course. This means he cannot somehow impart his momentum to the wall to make it move. But something that has a large momentum like a $5000 kg$ truck travelling at a velocity of about $70 km/hr$ could impart a momentum to the wall. Since the wall is fixed to the ground (i.e., it is not designed to move) it will break.

Thus we have the fundamentals of momentum. Now we invoke the concept of force. Force is defined as something that could change the state of motion of a body. A body at rest or having constant motion is said to be in the state of inertia. Inertia is the tendency of a body to continue in it's state, whatever be it is, while force is something that tends to change that state. So inertia of a body tells you the amount of force it could oppose. For example, if you push a ball with your hand, it will move. If you push with the same amount of force on a large massive cart, it will not move. So larger objects have a greater tendency to resist force. Otherwise speaking, large force is required to make a massive object move. So mass is a measure of inertia. Hence something that changes the state of inertia (force) should also depend on mass. So force is proportional to mass. Now, when you apply a force the momentum of the object changes (since the velocity of the object changes). Hence force is also proportional to change in velocity of the object. Higher the change in velocity, greater will be the force. Hence we say the force is proportional to change in momentum with respect to time.

So, we write

$$F\propto change\space in\space momentum\space w.r.t \space time=\frac{\vec{p_2}-\vec{p_1}}{t_2-t_1}$$ $$F\propto \frac{(m\vec{v_2}-m\vec{v_1})}{t_2-t_1}$$

where $\vec{v_1}$ is the initial velocity (velocity just before the force is applied) and $\vec{v_2}$ is the final velocity (the velocity just before the force is withdrawn). The time interval ($t_2-t_1$) indicates the time up to which the force is applied. Mass is not changing with time. The only quantity that varies with time on applying a force is the momentum, which means the velocity is alone changing with time. The change in velocity with respect to time is acceleration.

So, $$F\propto \frac{m(\vec{v_2}-\vec{v_1})}{(t_2-t_1)}=m\vec{a}$$

where $\vec{a}$ is the acceleration of the body. (We have taken the constant of proportionality as unity for consistency in the units). It is also a vector quantity. So, the force is mass times acceleration of a body. Since the momentum depends on mass, force also depends on mass, as clearly understood from the above examples. Also, if the velocity of a body decreases on applying a force (i.e., $\vec{v_2}<\vec{v_1}$), the acceleration will be negative. In such a case, the force opposes the motion. Now, if the applied force increases the velocity of the body (i.e., $\vec{v_2}>\vec{v_1}$), then the acceleration is positive and so the force favors the motion of the body. Whatever be the case, the force always points in the direction of change in momentum. That can be understood by analyzing in which direction the object is accelerating.

Now, if the force applied is zero, the acceleration will be zero, as on withdrawing the applied force, the mass will not vanish. Velocity is not changing with time. That means, velocity is a constant of motion and hence the body moves with constant velocity. Since the velocity is constant, the momentum is also constant. So, if no external force acts on a body, then the total linear momentum will be a constant. This statement is known as the law of conservation of momentum.

In that case, if there are no external force on a system of dynamical objects, then the total momentum before an event will be total momentum after an event. i.e., if we have two objects in our isolated system, then the sum of the momenta of the two bodies at any time will be a constant value. The individual value could change, but the total value is a constant.

• Does energy impart momentum or momentum impart energy to a system? Another question is that you say for Newton law,"We have taken the constant of proportionality as unity for consistency in the units".But why we do so in this case but not other (like "F" proportional to m1×m2/r^2 but we can't write directly F=m1×m2/r^2 instead we write as it proportional that is F=(constant)×(m1×m2/r^2))? – Abu sayed Jun 6 '16 at 4:10
• Energy causes motion. The observable momentum is a direct consequence of kinetic energy. So energy transfer take place by the mere process of contact. The fact is that both energy and momentum are getting transferred. Momentum is a measurable realization of kinetic energy. We cannot distinguish the terms and speak their solo properties. By collision two body are made in contact with a force that transfers energy from one to other. This we see as momentum. – UKH Jun 6 '16 at 9:51
• One Newton is equal to one kgm/s2. We defined one Newton as above and so the constant of proportionality is 1. In the case of law of gravitation the presence of G along with a mass and the inverse square of distance makes the quantity equal to acceleration. – UKH Jun 6 '16 at 9:55
• But how we define 1 kg of something called "Mass" as Newton wrongly stated "quantity of matter" which is quantity of 'number' concept replaced by "quantity of inertia"as Euler stated later. – Abu sayed Jun 17 '16 at 3:27

I will try my hand in simplicity...

Momentum is the measure of motion.

So, it is a measure of how much stuff is moving, and how fast that stuff is moving.

What we call mass at the very basic level is "amount of stuff". Very interestingly, more stuff is heavier (pulled more strongly by the Earth - gravitational mass) and harder to move around (change velocity - inertial mass). By some conspiracy of nature (which we think general relativity explains) these are the same to the best of our knowledge. So, more stuff is more stuff, and we call the answer to "how much stuff" the mass.

This all depends on experiment, you see. By experimenting, we have observed that two pieces of stuff moving at a certain speed (let us consider one dimension here, which is sufficient to understand what momentum is) can be exchanged for one piece of stuff (that is, half the mass) which was originally at rest moving double that speed, with the original two pieces of stuff having stopped. You have to arrange the collisions carefully, and avoid affecting the three pieces externally, but this can be done.

So, observation of colliding masses (which is essentially electromagnetic interaction, but let us not digress) with negligible external effects (like friction) you can arrive at the following conclusions:

(1) The amount of stuff moving can change.

(2) The speed of motion can change.

So that is not really a lot. But as it turns out, the product of stuff and speed summed together does not change. This can be confirmed by further experimentation - so we give it a name. Call it momentum. (We could have called it Mickey Mouse. Not convenient, but possible. A name is a name.)

In trying to describe how this quantity, which is conserved but can be exchanged, is actually exchanged, we introduce the concept of forces. A force is how momentum exchanges hands, so to speak. How much momentum is exchanged depends on how much force is applied and for how much time it is applied. (This even holds relativistically, if you use the relatvisitic momentum.) If the force is internal, that is between two pieces of the system you are studying, one gains momentum while the other loses it -- and in the exact same amount. This is essentially Newton's third law. But, if you apply an external force (whose reaction force, that is, canceling pair is not within your system) the momentum of the system you are studying will change. You can always expand your system and include more stuff, and ultimately you will have a system whose momentum is conserved. Thus, as far as we believe, the momentum of the universe is fixed.

Although not strictly part of the answer let me add the following:

Conservation of momentum is the coolest conservation law ever. It is always valid. Relativity? Yes (use the correct form). Quantum systems? Yes. Newtonian mechanics? Of course!

Energy is also very carefully conserved by nature, but it is a bit more subtle. When studying macroscopic objects, you need to track heat exchange and internal energy. Also, potential energy is sometimes easy to miss. Momentum? Very nice and simple.

... well, sometimes in charged particle collision, you get a radiated photon which you need to be careful about (it carries some momentum away). But that is significant when things are relativistic. Oh well.