Compare this to kinetic energy, which only has a magnitude. Kinetic energy is simply a single number scalar , but momentum has a magnitude in each direction of space. In other words, momentum can always be divided into its vector components which represent how much momentum an object has in each direction. The point is that for momentum, three different values its components are required to specify it.
On the other hand, only one value is needed to specify the kinetic energy its magnitude. One of the most obvious differences between kinetic energy and momentum is that kinetic energy depends quadratically on velocity it increases as v 2 , while momentum depends linearly on velocity it increases as just v. This means that kinetic energy actually increases way faster with velocity as momentum does. A nice way to visualize this is by graphing them both as shown below.
Kinetic energy, however, is not as straightforward and it may seem odd at first that it has a v 2 -term.
So, why exactly is kinetic energy proportional to velocity squared? Kinetic energy being proportional to velocity squared is simply a mathematical consequence of the work-energy theorem, which results from force being integrated over distance. If special relativity is accounted for, it also predicts that kinetic energy should be proportional to velocity squared. Then, we can make use of the fact that the distance dr is equal to velocity multiplied by time dt.
The point here is that assuming that the wok-energy theorem is true, it is inevitable that kinetic energy should depend on the square of velocity.
There is also another, arguably more fundamental way to see why momentum has a linear velocity-dependence and kinetic energy has a quadratic one and this comes from special relativity. I actually have a whole section later in the article discussing both momentum and kinetic energy in special relativity and in general relativity! The interesting thing is that this Lorentz factor can be expanded in something called a Taylor series which has infinitely many terms of increasing exponents as follows:.
From this, we can see an interesting fact; momentum contains terms with v, v 3 , v 5 and so on, while kinetic energy has terms with v 2 , v 4 , v 6 etc.
Now, the significance of this is that kinetic energy will always be positive, while momentum can be either positive or negative. This is because any even power of velocity has to be positive. An important distinction between momentum and kinetic energy is that, in certain situations, their conservation properties may be different.
Momentum is always a conserved quantity, while kinetic energy is not. Kinetic energy may be converted to other forms of energy, but total momentum in a system always remains the same.
Examples of this are collisions in which momentum is conserved, but kinetic energy gets converted into heat and sound. This really comes down to how we define these two quantities. Kinetic energy is only one form of energy other forms include mass and potential energy and the conservation of energy only applies to total energy.
Now, the total energy in any system will always be conserved this is the law of energy conservation! Therefore, it can always get converted to other forms of energy. Therefore, momentum is always conserved in a given system it will always be conserved if you actually take into account the whole universe.
This is, however, a little more technical than it may first appear. Momentum actually comes in two forms: linear momentum and angular momentum. Angular momentum, on the other hand, is the momentum associated with rotating or spinning motion. The important thing is that these two forms of momenta cannot be converted into one another and they are also independently conserved.
So, the bottom line is that momentum is always conserved, while kinetic energy may not be. For example, if a system has a translational symmetry, it means that the total momentum will be conserved. Namely, momentum conservation is related to spacial translation and symmetries associated with them.
There is certainly a conservation law for total energy, but not specifically to kinetic energy. Now, in some cases, if the only form of energy an object has is kinetic energy and probably mass as well , then the conservation of total energy will correspond to kinetic energy being conserved.
In other words, there are certain situations where kinetic energy is conserved, but it is not necessarily always conserved while momentum, on the other hand, is always conserved. One of the most important differences between momentum and kinetic energy in terms of their applications can be seen in collisions. The distinction here comes from how the two quantities are conserved in collisions. Now, a collision in physics is simply a process in which two or more objects interact in such a way that they exert forces on one another.
Usually, a collision is defined as an event in which the objects exert forces on each other for a fairly short duration think of a car crash, for example. Collisions are typically classified as either elastic or inelastic based on how they conserve or change the kinetic energy of a system. Total momentum will always be conserved in all types of collisions as it is generally a conserved quantity. Generally, momentum is conserved in all collisions. Kinetic energy, however, is not conserved in all collisions, only in special cases called elastic collisions.
In the case of an inelastic collision, some kinetic energy is always converted to heat or sound, meaning that it is not conserved. In this case, the kinetic energy is conserved also. Inelastic collisions, on the other hand, are the opposite. They are defined as collisions in which kinetic energy DOES get converted to heat, sound or other forms of energy. In any case, momentum will always be conserved in all types of collision , as required by the law of momentum conservation.
Now, some confusion may sometimes arise from the fact that kinetic energy may not be conserved even though there does exist a conservation law for energy. So, while kinetic energy may not be conserved in a collision, if you were to theoretically measure and add up ALL the different forms of energy associated with a system heat, sound etc. Kinetic energy is conserved under the condition that the collision is elastic.
An elastic collision is a collision in which no kinetic energy is converted to heat, sound or any other forms of energy, meaning that the kinetic energy will remain the same it is conserved. Elastic collisions, however, are quite rare in the real world no real collision is ever perfectly elastic. Momentum, however, is conserved in all collisions and it just transfers between the colliding objects. In general, it is not possible for a collision to conserve kinetic energy without conserving momentum as the law of momentum conservation prohibits this.
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.
It helps to confuse the opponent. Answer: b. 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. Can increase but not decrease. We say that the potential energy is transformed into kinetic energy, which is then spent driving in the nail.
We should emphasize that both energy and work are measured in the same units, joules. In the example above, doing work by lifting just adds energy to a body, so-called potential energy, equal to the amount of work done. From the above discussion, a mass of m kilograms has a weight of mg newtons. It follows that the work needed to raise it through a height h meters is force x distance, that is, weight x height, or mgh joules. This is the potential energy. Historically, this was the way energy was stored to drive clocks.
Large weights were raised once a week and as they gradually fell, the released energy turned the wheels and, by a sequence of ingenious devices, kept the pendulum swinging. The problem was that this necessitated rather large clocks to get a sufficient vertical drop to store enough energy, so spring-driven clocks became more popular when they were developed.
A compressed spring is just another way of storing energy. It takes work to compress a spring, but apart from small frictional effects all that work is released as the spring uncoils or springs back. The stored energy in the compressed spring is often called elastic potential energy , as opposed to the gravitational potential energy of the raised weight. Kinetic energy is created when a force does work accelerating a mass and increases its speed.
Just as for potential energy, we can find the kinetic energy created by figuring out how much work the force does in speeding up the body. Remember that a force only does work if the body the force is acting on moves in the direction of the force.
For example, for a satellite going in a circular orbit around the earth, the force of gravity is constantly accelerating the body downwards, but it never gets any closer to sea level, it just swings around. Consider, in contrast, the work the force of gravity does on a stone that is simply dropped from a cliff. In one second, the stone will be moving at ten meters per second, and will have dropped five meters.
How does the kinetic energy increase with speed? Think about the situation after 2 seconds. The mass has now increased in speed to twenty meters per second. It has fallen a total distance of twenty meters average speed 10 meters per second x time elapsed of 2 seconds.
The essential point is that the speed increases linearly with time, but the work done by the constant gravitational force depends on how far the stone has dropped, and that goes as the square of the time. For stones of different masses, the kinetic energy at the same speed will be proportional to the mass since weight is proportional to mass, and the work done by gravity is proportional to the weight , so using the figures we worked out above for a one kilogram mass, we can conclude that for a mass of m kilograms moving at a speed v the kinetic energy must be:.
Exercises for the reader : both momentum and kinetic energy are in some sense measures of the amount of motion of a body. How do they differ? Suppose two lumps of clay of equal mass traveling in opposite directions at the same speed collide head-on and stick to each other. Is momentum conserved? Is kinetic energy conserved?
0コメント