Introduction to Physics The Language of the Universe

1. Algebraic Mechanics In elementary and middle school, we learn mathematic for the sake of mathematic, we are never told what mathematics can ultimately be used for or why mathematics is useful other than the fact that its can help us make change and do our taxes one day. What they should be telling you is that the laws of the universe are written in a language that people are capable of understanding and that this language is mathematics. 1. Algebraic Mechanics The incredible thing about describing the universe through mathematics is that it not only allows us to describe things we have seen but it also allows us to take those observations and project them into the future. An example of this ability to accurately forecast events is called projectile motion or rocket science (no big deal). A simple example of this is used to describe the path a cannon ball will take when fired out of a cannon. Consider the below illustration: If we are given the starting velocity of the ball and angle of the cannon it just takes some basic algebra/trigonometry to solve not only for the final resting position of the ball but also its position in space at any time following its release. This come about by using the fact that in physics we can treat any two orthogonal directions (at 90 degrees with each other) as independent. Therefore, we can write the equations for the x-position (shown above) and y-position (up in the picture) of the cannon ball and then we will know everything there is to know about the ball for all of time. For those interested these equations are: Where x/y is the position in the x/y-direction, v is the initial velocity in the x and y direction depending on the subscript, g is the acceleration of gravity which on earth is about 9.8 meters per second, and t is the time. The two velocities can be found using the sine (for y-direction) or cosine (for x) of the firing angle times the total velocity. Therefore, using these two simple equations we can describe the motion of a cannon ball through space (with just algebra). Adding mathematical tools like calculus and differential equations allows us to expand our methods of describing the universe beyond what most people would think is possible. 2. Conservation Laws – A Tool for Physicists Now given the last example it is easy to think, â€Å"well how did you come by these equations for the cannon ball and can we derive equations like this for a general problem?† The answer to this question is yes and how can be explained using every psychists favorite tool, conserved quantities. Formally a conserved quantity, let’s call it â€Å"qconst† is defined as: This is just calculus talk for saying that this quantity does not change with time. In other words, if we measure this thing (for example the number of jellybeans in a jar) then no matter when we measure it we will always get the same amount. This in incredibly useful because there are several quantities in nature that are guaranteed to ALWAYS be conserved. An example of one of these quantities is momentum which is defined as: An example of this being useful for physics is the case where two objects collide with each other. Say we have a pool ball sitting motionless on a pool table and you hit this ball with a second ball of the same size. The total momentum before and after the collision (since momentum is conserved) is therefore represented by the equation: Where v#start is the starting velocity for each ball, which as we mentioned before is zero for ball number 2 and v