# Work conservation of energy and momentum relationship

### Momentum Conservation Principle

Learn the difference between momentum and kinetic energy. This is just the nature of the physics at work which we arrive at using mathematics. If the stiffness . Work. Forms of Energy. ◦ Potential Energy. ◦ Kinetic Energy Law of Conservation of Momentum The equation to calculate potential energy. Relation between momentum and kinetic energy. Sometimes it's desirable to express the kinetic energy of a particle in terms of the momentum.

To get a more quantitative idea of how much work is being done, we need to have some units to measure work. This unit of force is called one newton as we discussed in an earlier lecture. Note that a one kilogram mass, when dropped, accelerates downwards at ten meters per second per second. This means that its weight, its gravitational attraction towards the earth, must be equal to ten newtons.

From this we can figure out that a one newton force equals the weight of grams, just less than a quarter of a pound, a stick of butter. The downward acceleration of a freely falling object, ten meters per second per second, is often written g for short. Now back to work.

### Relation between momentum and kinetic energy

In other words approximately lifting a stick of butter three feet. This unit of work is called one joule, in honor of an English brewer. To get some feeling for rate of work, consider walking upstairs. A typical step is eight inches, or one-fifth of a meter, so you will gain altitude at, say, two-fifths of a meter per second.

Your weight is, say put in your own weight here! A common English unit of power is the horsepower, which is watts. Energy Energy is the ability to do work. For example, it takes work to drive a nail into a piece of wood—a force has to push the nail a certain distance, against the resistance of the wood. A moving hammer, hitting the nail, can drive it in. A stationary hammer placed on the nail does nothing.

Another way to drive the nail in, if you have a good aim, might be to simply drop the hammer onto the nail from some suitable height. By the time the hammer reaches the nail, it will have kinetic energy. It has this energy, of course, because the force of gravity its weight accelerated it as it came down.

Work had to be done in the first place to lift the hammer to the height from which it was dropped onto the nail. In fact, the work done in the initial lifting, force x distance, is just the weight of the hammer multiplied by the distance it is raised, in joules. But this is exactly the same amount of work as gravity does on the hammer in speeding it up during its fall onto the nail.

Therefore, while the hammer is at the top, waiting to be dropped, it can be thought of as storing the work that was done in lifting it, which is ready to be released at any time. To give an example, suppose we have a hammer of mass 2 kg, and we lift it up through 5 meters. This joules is now stored ready for use, that is, it is potential energy. 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.

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.

## Momentum & Energy

The way to understand this is to look at the mathematical representation of momentum and kinetic energy for a particle; where a particle is used as a reasonable representation for the objects. Therefore, the object weighing 0. In fact, the kinetic energy of the 0. The kinetic energy of an object is directly proportional to how much damage it will inflict when it strikes something.

But interestingly enough, it's the object weighing less that will do more damage, for a given momentum. And it is the square of the velocity in the kinetic energy equation which makes this effect possible. You can generalize this result to say that the ratio of kinetic energy of two objects having the same momentum is the inverse ratio of their masses.

This principle readily applies to firearms. When a bullet leaves a gun it has the same momentum as the gun which recoilsdue to conservation of momentum. But the bullet has much more kinetic energy than the gun.