Understanding Momentum in Physics

Understanding Momentum in Physics Momentum is a derived quantity, calculated by multiplying the mass, m (a scalar quantity) times velocity, v (a vector quantity). This means that the momentum has a direction and that direction is always the same direction as the velocity of an objects motion. The variable used to represent momentum is p. The equation to calculate momentum is shown below. Equation for Momentum:p mv The SI units of momentum are kilograms * meters per second, or kg*m/s. Vector Components and Momentum As a vector quantity, momentum can be broken down into component vectors. When you are looking at a situation on a 3-dimensional coordinate grid with directions labeled x, y, and z, for example, you can talk about the component of momentum that goes in each of these three directions: px mvxpy mvypz mvz These component vectors can then be re-constituted together using the techniques of vector mathematics, which includes a basic understanding of trigonometry. Without going into the trig specifics, the basic vector equations are shown below: p px py pz mvx mvy mvz Conservation of Momentum One of the important properties of momentum and the reason its so important in doing physics is that it is a conserved quantity. That is to say that the total momentum of a system will always stay the same, no matter what changes the system goes through (as long as new momentum-carrying objects are not introduced, that is). The reason that this is so important is that it allows physicists to make measurements of the system before and after the systems change and make conclusions about it without having to actually know every specific detail of the collision itself. Consider a classic example of two billiard balls colliding together. (This type of collision is called an elastic collision.) One might think that to figure out whats going to happen after the collision, a physicist will have to carefully study the specific events that take place during the collision. This actually isnt the case. Instead, you can calculate the momentum of the two balls before the collision (p1i and p2i, where the i stands for initial). The sum of these is the total momentum of the system (lets call it pT, where T stands for total) and after the collision- the total momentum will be equal to this, and vice versa. (The momenta of the two balls after the collision is p1f and p1f, where the f stands for final.) This results in the equation: Equation for Elastic Collision:pT p1i p2i p1f p1f If you know some of these momentum vectors, you can use those to calculate the missing values, and construct the situation. In a basic example, if you know that ball 1 was at rest (p1i 0) and you measure the velocities of the balls after the collision and use that to calculate their momentum vectors, p1f and p2f, you can use these three values to determine exactly the momentum p2i must have been. (You can also use this to determine the velocity of the second ball prior to the collision, since p / m v.) Another type of collision is called an inelastic collision, and these are characterized by the fact that kinetic energy is lost during the collision (usually in the form of heat and sound). In these collisions, however, momentum is conserved, so the total momentum after the collision equals the total momentum, just as in an elastic collision: Equation for Inelastic Collision:pT p1i p2i p1f p1f When the collision results in the two objects sticking together, it is called a perfectly inelastic collision, because the maximum amount of kinetic energy has been lost. A classic example of this is firing a bullet into a block of wood. The bullet stops in the wood and the two objects that were moving now become a single object. The resulting equation is: Equation for a Perfectly Inelastic Collision:m1v1i m2v2i (m1 m2)vf Like with the earlier collisions, this modified equation allows you to use some of these quantities to calculate the other ones. You can, therefore, shoot the block of wood, measure the velocity at which it moves when being shot, and then calculate the momentum (and therefore velocity) at which the bullet was moving prior to the collision. Momentum and the Second Law of Motion Newtons Second Law of Motion tells us that the sum of all forces (well call this Fsum, though the usual notation involves the Greek letter sigma) acting on an object equal the mass times acceleration of the object. Acceleration is the rate of change of velocity. This is the derivative of velocity with respect to time, or dv/dt, in calculus terms. Using some basic calculus, we get: Fsum ma m * dv/dt d(mv)/dt dp/dt In other words, the sum of the forces acting on an object is the derivative of the momentum with respect to time. Together with the conservation laws described earlier, this provides a powerful tool for calculating the forces acting on a system. In fact, you can use the above equation to derive the conservation laws discussed earlier. In a closed system, the total forces acting on the system will be zero (Fsum 0), and that means that dPsum/dt 0. In other words, the total of all momentum within the system will not change over time which means that the total momentum Psum must remain constant. Thats the conservation of momentum!

Dawned vs. Donned

Dawned vs. Donned Dawned vs. Donned Dawned vs. Donned By Guest Author This is a guest post by Shelley DuPont. If you want to write for Daily Writing Tips check the guidelines here.    Did you ever think you knew the lyrics to a song, only to find out later they were nothing like you thought?   I know I have. Editor’s note: There’s a word for this kind of mishearing: mondegreen, â€Å"a misunderstood or misinterpreted word or phrase resulting from a mishearing, esp. of the lyrics to a song.† Sometimes a writer who has misunderstood an expression passes it on to the reader, with unintended results. For example, I came across a guest post on how conversations can lead indirectly to gaining more business for yourself.   Anticipating a great article, I eagerly started reading. Right at the end of the introductory paragraph, I saw it. It hit me like a small flaw on an expensive piece of clothing:   It donned on me The context called for it dawned on me, meaning I understood, I became aware of. Wanting to reassure myself that I hadnt been mistaken in my own understanding of the idiom, I googled the phrase as the author used it.   There it was at the top of the page. Google was asking, Do you mean It dawned on me?' Yes! Yes! I said with some relief.   But this still didnt answer the question as to why the author chose to use donned as the verb.    Used figuratively, the verb to dawn means â€Å"to begin to appear or become visible† in the sense of mental enlightenment or awareness.   If something dawns on you, then a new understanding has come your way.   This use seems fairly obvious, especially within the context of the article.   The verb to don, on the other hand, means to put on clothing, or, in a figurative sense, to assume, or to get into.   For example, a recent headline taken from the New York Times entertainment page reads, Amanda Seyfried likely to don â€Å"Red Riding Hood† I suppose the author could have done it deliberately. I did discover a website named itdonnedonme that focuses on competitive 24 hour film making. But the blog title is a deliberate pun on the expression it dawned on me and the blog owner’s name: Evan Donn. The context of the article I was trying to read definitely called for â€Å"dawned.† Maybe the error arose from the writer’s pronunciation of the words dawn and don. dawn [dà ´n] rhymes with lawn, yawn and aw (as in â€Å"Aw, shucks!) don [dÃ… n] rhymes with on, con and Ron At this point, I can only conclude that the author, like those of us singing the wrong words for years, has simply mixed up â€Å"dawn† and â€Å"don.†   By the way, it just dawned on me that I never did finish reading that article. Shelley DuPont is a former high school English teacher who blogs for local business owners. She actively tutors online ESL students, freelance writes, and paints. Want to improve your English in five minutes a day? Get a subscription and start receiving our writing tips and exercises daily! Keep learning! Browse the Misused Words category, check our popular posts, or choose a related post below:Avoid Beginning a Sentence with â€Å"With†Story Writing 101Double Possessive