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?
[Physics] actually meant by momentum at the deepest level of physics
momentumnewtonian-mechanics
Related Solutions
$F = \frac{dp}{dt}$ means that force is the rate of momentum transfer per unit time.
Lets say we have mass $m_1$ moving to the right, and mass $m_2$ is on the left side of $m_1$ with zero velocity. If $m_1$ put a force to pull $m_2$, that force will create the acceleration on $m_2$ and increase its velocity, this also means the change in momentum. At the same time the reaction force will also slow down the mass $m_1$ and decrease its momentum. It you think of it that way, you can see that the force between these two masses is just the rate of transfer of momentum from $m_1$ to $m_2$.
$$ F = ma = m\frac{dv}{dt} = \frac{d(mv)}{dt} = \frac{dp}{dt} $$
Don't be surprised that physics has a lot of definitions that are circular. Ultimately, we are just describing the universe.
Work and energy have been defined in a certain way in newtonian physics to explain a kinematic model of reality. This is a model, not reality - you will find no such thing in reality. However, in many scenarios, it is close enough to reality to be useful.
For example, let's say that a human has a 10% efficiency at converting food to mechanical work. So if you spend 1000 kJ of food energy to press against a wall, are you doing 1000 kJ of work, or 100 kJ of work, or 0 kJ of work?
In strict mechanical sense, you did no work whatsoever, and all of the energy you used was wasted as heat. If you instead used this energy to push a locomotive, you would have wasted "only" 900 kJ of the energy as heat, with 100 kJ being work. But the locomotive has its own friction, and it wil stop eventually, wasting all the energy as heat again. And overall, you did expend all those 1000 kJ of food energy that is never coming back.
All of those are simplifications. Kinematics is concerned with things that move. Using models is all about understanding the limits of such models. You're trying to explain thermodynamics using kinematics - this is actually quite possible (e.g. the kinematic theory of heat), but not quite as simple as you make it. Let's look at the fire example. You say there is no displacement, and therefore no work. Now, within the usual context kinematics is used, you are entirely correct - all of that energy is wasted, and you should have used it to drive a piston or something to change it to useful work.
Make a clear note here: what is useful work is entirely a human concept - it's all 100% relevant only within the context of your goals; if you used that "waste" to heat your house, it would have been useful work. It so happens that if you look closer, you'll see that the heat from the fire does produce movement. Individual molecules forming the wood wiggle more and more, some of them breaking free and reforming, and rising with the hot air away from the fire, while also drawing in colder air from the surroundings to feed the fire further. There's a lot of displacement - individual molecules accelerate and slow down, move and bounce around... But make no mistake, the fact that kinematics can satisfactorily explain a huge part of thermodynamics is just a bonus - nobody claimed that kinematics explains 100% of the universe. It was a model to explain how macroscopic objects move in everyday scenarios. It didn't try to explain fire.
For your specific questions, you really shouldn't ask multiple questions in one question. It gets very messy. But to address them quickly:
- There is no potential energy in the kinematic model. The concept is defined for bound states, which do not really exist as a concept in kinematics. In other models, you might see that there's a difference between, say, potential energy and kinetic energy - no such thing really exists in reality. You need to understand the context of the model.
- In a perfectly kinematic world, this is 100% correct. However, as noted before, kinematics isn't a 100% accurate description of reality, and there are other considerations that apply, such as the fact that humans have limited work rate, limited ability to apply force, and the materials we are built of aren't infinitely tough, perfectly inflexible and don't exist in perfect isolation from all the outside (and inside) effects. In real world applications of models, these differences are usually eliminated through understanding the limits of given models, and using various "fixup" constants - and if that isn't good enough, picking (or making) a better model.
- You're mixing up many different models at different levels of abstraction and of different scope so confusion is inevitable. Within the simplified context of kinetics, there is no concept of "potential energy". You simply have energy that can be used to do work, and that's it; it doesn't care about how that energy is used to do work, about the efficiency of doing so etc. In another context, it might be very useful to think of energy and mass as being the same thing - and in yet another, they might be considered interchangeable at a certain ratio, or perhaps in a certain direction, or at a certain rate. It's all about what you're trying to do.
- How is that equation useful? That's the only thing that matters about both definitions and equations. I can define a million things that are completely useless if I wanted to - but what's the point?
- Within the original context, those aren't considered at all. Within a more realistic context, both heat and sound are also kinematic.
The reason you have so much trouble finding the answer to your questions on physics sites and forums is that the question doesn't make much sense in physics. It's more about the philosophy of science, and the idea of building models of the world that try to describe reality to an approximation that happens to be useful to us. You think that those words have an inherent meaning that is applicable in any possible context - this simply isn't true. From the very inception of the idea of physics, people have known that it isn't (and never will be) an accurate representation of reality; and we've known for a very long time that, for example, different observers may disagree on the energy of one object. You just need to understand where a given model is useful, and pick the right model for the job. Don't try to drive a screw with a garden rake.
Best Answer
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.