What is the difference between momentum and energy

An object placed in a resting posting has. Potential energy. Kinetic energy. The total energy in an object including rest energy in the world.

Cannot change. Can decrease but not increase. May either decrease or increase. Connect and share knowledge within a single location that is structured and easy to search. Now my problem is the following: Suppose you want to explain someone without mentioning the formulas what's momentum and what's kinetic energy. How to do that such that it becomes clear what's the difference between those two quantities?

I think is is relatively easy to explain the first two points using everyday language, without referring to formulas. If you shoot a bullet, the rifle recoils with the same momentum as the bullet, but the bullet has a lot more Kinetic energy. Aren't you glad your shoulder is being hit by the rifle stock, and not by the bullet? As a matter of fact, the second point couldn't be more wrong.

Momentum is far more ubiquitous than kinetic energy since it is a conserved quantity of every physical system that is translationally invariant. With respect to your question, user Gerard gave an explanation as intuitive as it gets here.

Consider two equal size lumps of wet clay moving toward each other at the same speed. Things are moving. But the center of mass is not.

After they smush into each other, the resultant lump is not moving. Two lumps move in almost the same direction. They tap each other, and and the resultant lump keeps going. It seems we need two different ways of measuring motion to make sense of this.

The motion of the parts is much the same, but the total is very different. Momentum might be loosely defined as the quantify of motion. If two objects move at the same speed, the more massive one has a bigger quantity of motion. Of two equal mass objects, the faster one has more motion. Momentum is a vector. It has a direction and magnitude.

It might be tempting to think that they are fairly similar concepts, but in fact they are not. They are related concepts, but are distinct in the physical sense. To get you thinking more deeply about this imagine getting struck by two fast moving objects having the same momentum. One object weighs 10 kg and the other object weighs 0. Which object would inflict more damage? The intuitive response to this might be that they would both inflict an equal amount of damage given that they both have an equal amount of momentum.

As it turns out, the 0. This certainly does work, however, it is not quite correct. The problem is that this only deals with magnitudes of velocity and momentum , but in reality, these are both vector quantities. The correct thing to do is a vector derivative derivative of kinetic energy with respect to the velocity vector. Technically, the correct notation should also use a partial derivative , similarly as to how a gradient also uses partial derivatives.

In physics, derivatives with respect to vectors vector derivatives are very common and in fact, also the commonly used gradient is a form of a vector derivative. Now, what does taking a derivative with respect to a vector actually mean? It simply means that we take the derivative with respect to each component individually. This then produces another vector quantity:.

This dot product, on the other hand, can be calculated as a dot product is simply the sum of the products of these vector components :. Moreover, this is clearly just momentum, so we can then say that:. The interesting thing about momentum as the derivative of kinetic energy is what it physically represents.

Kinetic energy, at a high level, is the energy associated with motion. Momentum can then be thought of as a measure of how much this kinetic energy changes in a particular direction. This also helps to explain why momentum is a vector and kinetic energy a scalar and how exactly they differ. Think of it this way: if the velocity of an object changes, say in the x-direction, then the derivative of kinetic energy with respect to the x-velocity gives you the momentum in the x-direction.

Therefore, physically, a positive x-component of the momentum represents an increase in kinetic energy due to an increase in velocity in the x-direction and vice versa a negative x-momentum represents a decrease in kinetic energy. In Lagrangian mechanics, this basically works as a definition for momentum. If momentum is the derivative of kinetic energy, does this mean that kinetic energy is then the integral of momentum?

In short, kinetic energy is indeed the integral of momentum with respect to velocity. More accurately, the line integral of momentum with respect to the velocity vector at each point along a path gives the total change in kinetic energy. Physically, this represents the work done along the path. Technically, this is actually a line integral , which simply means an integral along a specific path. Now, the physical meaning of this is that by adding up integrating all the momenta at each point along a path described by a certain velocity at each point, we can calculate the total change in kinetic energy between the end points of the path see the picture below.

You can see this quite easily by simply writing out the dot product and splitting out the integrals component by component:. An interesting extra fact is that velocity can actually be expressed as the derivative of kinetic energy with respect to momentum. All we do for this is use the following formula for kinetic energy in terms of momentum:. However, in more advanced classical mechanics, the same concepts motion, dynamics etc. This is what Lagrangian mechanics is all about.

In Lagrangian mechanics, everything is described by energies and it is always done by defining a Lagrangian for any given system. Generally, for any system, the Lagrangian is the difference between the kinetic and potential energies of each object in the system:.

Momentum, on the other hand, has a different definition. Momentum in Lagrangian mechanics is defined as the derivative of the Lagrangian with respect to velocity :. This is actually not quite the correct definition yet, but from this, you can clearly see the similarity to what I explained earlier about momentum as the derivative of kinetic energy.

By using these, the definition for momentum is a little different, but the basic idea is the same. You can read more about this as well in my introductory article link above. Anyway, in most cases, the above definition is simply just momentum equals the derivative of kinetic energy, however, not always. This is a perfect example of how momentum is defined differently in Lagrangian mechanics the Lagrangian definition is actually much more fundamental! You should be able to understand it with basic high school level math and physics.

In special relativity, both momentum and kinetic energy are defined a little differently and the special relativistic formulas are valid at high velocities close to the speed of light , while the usual classical formulas are not.

The key point in special relativity is that all the equations are slight corrections to classical mechanics which only come to play at high velocities. Anyway, the goal of this section is to look at the relationship between momentum and kinetic energy in special relativity. It is possible to derive the following expression for kinetic energy in terms of momentum:. In special relativity, there is also an analogous relation of momentum as the derivative of kinetic energy.

Based on this, we can also write kinetic energy as the integral of momentum as follows more accurately, the change in kinetic energy :. Both of these equations are quite easy to verify if you simply know how to take derivatives and integrals. General relativity is a theory of gravity that incorporates the ideas of special relativity spacetime, four-vectors and whatever else. In general relativity, it is actually not possible to define a useful notion of kinetic energy.