This can, however, occur the other way around, meaning that a collision can conserve momentum without conserving kinetic energy. The collisions in which momentum is conserved but kinetic energy is not are again called inelastic collisions. However, there does not exist a collision type in which momentum would not be conserved. In some cases, kinetic energy can increase after a collision and this is called a super-elastic collision. In this type of collision, some potential or chemical energy of the objects involved is converted into kinetic energy, meaning that the total kinetic energy is actually higher after the collision.
If the other ball has a nitroglycerin core a very realistic example, I know! In this case, the explosion a chemical reaction would release a huge amount of chemical energy that would be converted into the kinetic energy of the other ball the explosion would cause the other ball flying off with a much greater velocity than before the collision. This would be an example of a super-elastic collision. Super-elastic collisions are even thought to occur naturally in some cases.
Examples of this are collisions between plasmoids caused by the Sun , according to a study published in the Nature Physics -journal.
Plasmoids are essentially clumps of plasma that have a specific shape due to a magnetic field. In the study mentioned above, the researchers essentially looked at coronal mass ejections caused by the Sun which are plasmoids. They then concluded that these plasmoids behaved in a way that was very similar if not identical to a super-elastic collision the kinetic energies of the plasmoids actually increased as they collided, just like how super-elastic collisions would work.
The derivation of this formula is extremely simple. From just that one formula, we could easily calculate either the momentum or the kinetic energy of an object if we know the other. Generally, the more momentum an object has, the more kinetic energy it will also have. This is because both momentum and kinetic energy depend on velocity, so if one increases, the other one will as well. It is possible, however, for momentum to be negative while kinetic energy is always positive.
This is, however, only true for the magnitude of the momentum. If momentum were to be negative and increase in the negative direction i. Now, momentum being negative is somewhat arbitrary since it only depends on which way we choose the positive velocity -direction to be. In this case, the velocity and momentum are negative. This is obviously a positive number.
Generally, kinetic energy is always positive because it contains a v 2 -term and the square of a real number is always positive. In short, momentum is proportional to the square root of kinetic energy since momentum is directly proportional to velocity, while kinetic energy is proportional to velocity squared. This means that if kinetic energy becomes four times as big, the momentum will only double. This can also be seen from the relationship between momentum and kinetic energy if we solve for p:.
A single object cannot have any kinetic energy if it has no momentum. This is because an object with no momentum implies it has no velocity either, which also means no kinetic energy. However, a system of multiple objects can have kinetic energy but no momentum in total. This can be seen from the example given below. For a single object, zero momentum always means zero kinetic energy as well. However, for systems of multiple objects, it is possible to have zero total momentum but non-zero kinetic energy in the case that the momenta of each object is exactly equal but in opposite directions.
If the momentum of this object is zero, the kinetic energy will be as well. However, the case with multiple objects is easiest to understand through an example. This means that they have equal but opposite momentum :. In this case, the kinetic energy will be the sum of the kinetic energies of each object :. This is clearly non-zero and we therefore have an example of a case where the total momentum is zero, but kinetic energy is not.
When I first got started learning calculus, the first thing I noticed was that the equations for momentum and kinetic energy looked very similar. So, is momentum actually the derivative of kinetic energy and why?
In short, momentum is the derivative of kinetic energy with respect to velocity as it describes the directional change in kinetic energy as the velocity changes. The derivative of kinetic energy with respect to velocity produces a vector quantity momentum , similarly to a gradient of a scalar function.
The key thing here is the fact that momentum is a vector while kinetic energy is a scalar. Naively, when I first noticed this, my first thought was to just take the derivative of kinetic energy with respect to velocity like this:. 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. In some special cases, it may be possible, but generally not. Maybe you can begin to see the problem here. We cannot define a notion of kinetic energy simply because of the complexity of these terms.
The average velocity of an object is the difference between final and the initial velocity in separate three dimensions divided by the total time. The velocity of an object is directly related to the kinetic energy of the object. Using classical mechanics the kinetic energy of an object is half times mass multiplied by velocity squared divided.
The theory of relativity suggests a more advanced version, which is not discussed here. The theory of relativity also suggests that the observed mass of an object increases when the velocity of the object is increased. The velocity of an object is dependent only on the changes of space time coordinate of the object. Momentum is a very important property of a moving object.
The momentum of an object is equal to the mass of the object multiplied by the velocity of the object. Since mass is a scalar, the momentum is a vector, which has the same direction as the velocity.
It states that the net force acting on an object is equal to the rate of change of momentum. Since mass is constant, on non-relativistic mechanics, the rate of change of momentum is equal to mass multiplied by the acceleration of the object. The most important derivation from this law is the momentum conservation theory.
This states that if the net force on a system is zero the total momentum of the system remains constant. Momentum is conserved even in relativistic scales.
It must be noted that the momentum is dependent on both the mass of the object and the space time coordinate change of the object.
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