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.
If an object has position x1 at time t1 and position x2 at time t2, the average velocity during that interval is
If the times t1 and t2 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/second2 (m/s2). 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/s2.
We compute acceleration in exactly the same fashion as we did velocity. If an object has velocity v1 at time t1 and velocity v2 at time t2, the average acceleration during that interval is
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.
When you have decided on your answer, click on the "Check Answer" button to find out if you did the problem
correctly. You may try the same type of problem with a different graph by clicking on the "Same Problem" button,
or choose to practice with either the next or previous type in sequence ("Next Problem" or "Previous Problem" buttons), or a random type of problem ("Random Problem" button). These control buttons are a common feature on the applets in this text.
What kind of gadget could this applet be describing? Think of the toys often found in dentists' offices to help small children occupy their time in the
waiting room: specifically the one with colored wooden balls that ride on wiggly wires. The graph in the problem
describes a one-dimensional problem, not unlike any particular wire on one of those toys. Imagine a long straight
wire with a red ball at some position. As a child moves it back and forth on the wire, the graph gives its position on
the wire as a function of time. After a while, an adult puts a black ball on the wire at exactly the place the
child started with the red one, and the child (miraculously!) moves it with precisely the same velocities and
accelerations as she did with the first one.
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 * 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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Kinematics
v = (x2 - x1) / (t2 - t1).
On 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.
a = (v2 - v1) / (t2 - t1).
On 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.
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 (which 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.
As before, if the times t1 and t2 are very close, the average acceleration becomes an instantaneous acceleration. If the velocity is decreasing, the acceleration will be negative; we do not usually use the term deceleration (except in the demonstration below).
A Graphical Example
3 * 9 * 9 * 9 * 9 * 9 * 9 * 9 * 9 * 9 * 9! / (2! 7!)
The "!" in the first expression denotes the factorial; n! is defined as the product of all the integers
less than or equal to n, and 0! is defined as 1.