# Mass and potential energy relationship to velocity

### Potential energy - Wikipedia If an object is dropped from height h = m, then the velocity just before impact is v = m/s. If the mass is m = kg, then the kinetic energy just before impact is equal to. In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potential energy of an object that depends on its mass and . In this section the relationship between work and potential energy is. Show that the gravitational potential energy of an object of mass m straight up at constant speed, then the force needed to lift it is equal to its weight \text{mg}. . The equation for change in potential energy states that {\text{ΔPE}}_{\text{g}}.

Typically, the ground is considered to be a position of zero height. But this is merely an arbitrarily assigned position that most people agree upon. Since many of our labs are done on tabletops, it is often customary to assign the tabletop to be the zero height position. Again this is merely arbitrary. If the tabletop is the zero position, then the potential energy of an object is based upon its height relative to the tabletop. For example, a pendulum bob swinging to and from above the tabletop has a potential energy that can be measured based on its height above the tabletop.

By measuring the mass of the bob and the height of the bob above the tabletop, the potential energy of the bob can be determined. Since the gravitational potential energy of an object is directly proportional to its height above the zero position, a doubling of the height will result in a doubling of the gravitational potential energy. A tripling of the height will result in a tripling of the gravitational potential energy. Use this principle to determine the blanks in the following diagram. Knowing that the potential energy at the top of the tall platform is 50 J, what is the potential energy at the other positions shown on the stair steps and the incline?

Elastic Potential Energy The second form of potential energy that we will discuss is elastic potential energy. Elastic potential energy is the energy stored in elastic materials as the result of their stretching or compressing. Elastic potential energy can be stored in rubber bands, bungee chords, trampolines, springs, an arrow drawn into a bow, etc.

The amount of elastic potential energy stored in such a device is related to the amount of stretch of the device - the more stretch, the more stored energy. Springs are a special instance of a device that can store elastic potential energy due to either compression or stretching. A force is required to compress a spring; the more compression there is, the more force that is required to compress it further. For certain springs, the amount of force is directly proportional to the amount of stretch or compression x ; the constant of proportionality is known as the spring constant k. If a spring is not stretched or compressed, then there is no elastic potential energy stored in it.

The spring is said to be at its equilibrium position. Now, the momentum is mv, mass x velocity. This means for an object having constant mass which is almost always the case, of course! Now think of a collision, or any kind of interaction, between two objects A and B, say. In other words, since these are vectors, they are of equal length but pointing in opposite directions. This means that for every bit of momentum A gains, B gains the negative of that.

In other words, B loses momentum at exactly the rate A gains momentum so their total momentum remains the same.

Relativistic Mass and Energy

But this is true throughout the interaction process, from beginning to end. Therefore, the total momentum at the end must be what it was at the beginning.

You may be thinking at this point: Nevertheless, we do know that momentum will be conserved anyway, so if, for example, the two objects stick together, and no bits fly off, we can find their final velocity just from momentum conservation, without knowing any details of the collision.

First, it only refers to physical work, of course, and second, something has to be accomplished. Consider lifting the box of books to a high shelf.

If you lift the box at a steady speed, the force you are exerting is just balancing off gravity, the weight of the box, otherwise the box would be accelerating. Putting these together, the definition of work is: 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.

### Kinetic and Potential Energy

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. In other words approximately lifting a stick of butter three feet.

## Potential Energy

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.

### Potential Energy

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.