**Mechanics** is the study of objects in motion, and sets the stage for our entire study of physics. We will begin by learning how to describe the motion of objects using mathematics; this is called **kinematics**. We will then learn about the causes of motion as we study **dynamics**. Finally, we will learn about **conserved quantities**, which do not change during the course of the motion. These will allow us to apply conservation principles to attain a deeper understanding of the mechanics of a problem. We will concern ourselves primarily with motion in a straight line, motion in a plane, and rotational motion about a fixed axis.

Since all motion involves a change of position over time, position and time will be two of our most important variables. **Position** is usually denoted by the letters "x" or "y", depending on whether the motion is horizontal or vertical. It is measured as a distance from a fixed point, called the **origin**. Position, therefore, always has units of distance; typically we will measure distance in units of meters, using the **Systeme Internationale (S.I.)** standard. We will typically measure **time**, denoted by the letter "t", in seconds. When we describe motion graphically, position will be the dependent variable (measured on the vertical axis) and time will be the independent variable (measured on a horizontal axis).

When discussing motion, it is natural to use the concept of the **rate of change** of a quantity: how much the quantity has changed during an interval of time, divided by the time interval. We define the **velocity** as the rate of change of position. Velocity is denoted by the letter "v", and has S.I. units of meters/second (abbreviated m/s). A meter per second is about 2.25 miles per hour.

v = (xOn a graph of position vs. time, this should be familiar to you as the "rise over the run": the slope. So on a graph of position as a function of time, the velocity at a point on the path of an object's motion is the slope of the line at that time._{2}- x_{1}) / (t_{2}- t_{1}).

If the times t_{1} and t_{2} are very close, the average velocity becomes an instantaneous velocity. We will assume that we are working with instantaneous quantities in this book, unless otherwise noted. Note that if the position is decreasing (moving left on the real number line), the velocity will be negative: velocities depend on our choice of coordinate system.

We define the **acceleration** as the rate of change of velocity. It is denoted by the letter "a", and has S.I. units of meters/second/second, or meters/second^{2} (m/s^{2}). Whenever we work with a rate of change, we get another factor of time in the denominator. In general, we could also concern ourselves with the rate of change of acceleration, or the rate of change of the rate of change of acceleration, etc., but a proper treatment of that requires calculus. For our purposes it will suffice to assume that the acceleration is a constant, so that all subsequent rates of change are zero. A car that goes from 0 to 60 miles per hour in 13 seconds has an acceleration of about 2 m/s^{2}.

a = (vOn a graph of velocity vs. time, this is again the slope. So on a graph of velocity as a function of time, the acceleration at a point on the graph is the slope of the line there._{2}- v_{1}) / (t_{2}- t_{1}).

Acceleration can also be seen in a graph of position vs. time: at any point where the slope is not a constant, the graph is curved and the acceleration is nonzero. Since we restrict ourselves to situations involving constant acceleration (As before, if the times twhich includes constant zero acceleration!), we will see that when the graph is curved it will be parabolic, and the sign of the acceleration can be determined by the shape of the parabola: concave up corresponds to positive acceleration, and concave down corresponds to negative acceleration.

**Speed** is the absolute value of velocity; while position, velocity and acceleration all include the concept of direction, speed does not.

The graph below is part of an **applet**; a computer program embedded in the text you are studying. This text contains a number of applets, all intended to give you practice with problem solving in physics. This applet
is a demonstration of a car starting at rest, undergoing constant acceleration for a while,
then undergoing constant deceleration. You can plot the position (the vertical scale goes from 0 to
800 meters), the velocity (from -50 m/s to 50 m/s) or the acceleration. In each case the horizontal
axis goes from zero to 20 seconds. You may choose the acceleration, deceleration and the time at which you
hit the brakes. Verify that the acceleration is indeed the slope of the velocity graph, and that the velocity graph is the slope of the position graph.

The next applet creates graphical problems involving the kinematical concepts of speed, velocity and acceleration. The horizontal axis represents time, and the vertical axis represents position. The red line describes the position of two balls as a function of time. The red ball has traveled along the path described by the line, and the black ball is following it. The program will ask you for the sign of the speed, velocity or acceleration of either ball, or for the speed, velocity or acceleration of one with respect to the other. You can figure out the sign of the acceleration by noting the values of the velocity on either side of the ball; if the velocity is increasing, the acceleration must be positive; if it is decreasing, the acceleration must be negative. In any given time interval (drawn as one grid square), the acceleration will always have the same sign.

And how many different position graphs does that program know? This is a **combinatorial problem**.
The program which generates those graphs first chooses one of 3 initial positions.
Then, for each of the 9 time segments, it chooses one of 9 possible graphs: horizontal, rising or dropping linear (either
of which can be steep or shallow), or rising or dropping concave up or down. Finally, it chooses two time segments for
the positions of the red and black balls. This last is formally called a **combination**: the number of ways of
choosing k unordered choices from among n possible choices. They are unordered from the point of view of the program because
no matter which one is picked first and which is picked second, the earlier one in time gets the black ball and later one
gets the red ball. So the number of possible graphs is

3 * 9 * 9 * 9 * 9 * 9 * 9 * 9 * 9 * 9 * 9! / (2! 7!)The "!" in the first expression denotes the= 3 * 9 * 9 * 9 * 9 * 9 * 9 * 9 * 9 * 9 * 9 * 8 / 2

= 41,841,412,812

There are often combinatorial problems in physics. Our favorite is in particle physics. There, the problem is to add up the contributions from all possible particle interactions which can take place between the particle reactants and the particle products. The combinatorial problem there is interesting because many of the interactions are the same (since all particles of a given type are indistinguishable), and you are not allowed to count any of them more than once!

The next section continues with our discussion on kinematics as we construct kinematical equations.

©2008, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.

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