PHY 206 Chapter 2 Motion in One Dimension / Dr. Miner

Kinematics

The major portion of this semester is the study of mechanics, the science of motion and its causes. In this chapter and the next we examine kinematics, which is a description of motion without regard to its causes. The causes are explored in the subject of dynamics starting in Chapter 4. The current chapter considers straight-line motion, or one-dimensional motion. Three-dimensional motion is treated in the next chapter.

Displacement

Displacement is the measurement of the location of a point in space relative to the position of another point in space. It could be the location of an object relative to the origin of a coordinate system, or the position of an object that has moved relative relative to its initial location. In general displacement is a three-dimensional vector. In special cases it could be one-dimenstional as in the motion along a straight line.

Distance and Displacement

Displacement is a vector which relates, say, final position to initial position for an object. The distance covered by the object is the length of string placed exactly along the path of the object. This distance would be simply a magnitude, a scalar. For example, if a car is driven 100 miles in some direction, and then 100 miles back to its starting place, its total displacement is zero and its distance covered is 200 miles.

Speed and Velocity

We now define speed to be the distance travelled divided by the time it takes. The units could be meters/second, or miles/hour. Both the distance and the time are scalars, and thus the speed is a scalar. If rather than the distance travelled we use the vector displacement and divide by the time, then we get the vector velocity. A vector divided by a scalar is a vector. If a car is driven north for 100 miles in two hours, its average velocity is 50 miles per hour north, a vector (magnitude and direction). Its average speed is 50 miles per hour, a scalar (magnitude only, no direction - but could include a minus sign).

Average and Instantaneous

The speed calculated by dividing 100 miles by two hours to get a scalar 50 miles per hour is called the average speed. (Miles per hour has the same meaning as miles/hour.) The actual speed at any time during the trip could be higher or lower than 50 miles per hour. If you want to know the actual speed at a particular time, you want the instantaneous speed. It is calculated the same way with the exception that the time interval used must be so small that the speed cannot change noticeably in that time period. In the language of calculus, we need for the time interval to approach zero. If one thinks about a plot or graph of distance versus time, the average speed would be the average slope of the graph. The instantaneous speed at a particular time is the slope at that value of time. The time interval for the measurement of the instantaneous slope must be so small that you cannot tell the difference between the plotted line and a straight line.

Acceleration

We define acceleration in a manner similar to velocity. In one-dimensional motion the scalar acceleration is the difference in speed (after and before) divided by the time it takes. The units will be meters per second-squared, or m/s². In three-dimensional motion, the change in velocity will be a vector. When divided by the scalar time, the result is the vector acceleration.

Constant Acceleration Equations

Here we will stay with one-dimensional motion. We also assume that acceleration remains constant. Then using the definitions of velocity and acceleration we can establish the three Constant Acceleration Equations as follows:

v = v_o + a t

x = x_o + v_o t + ½ a t²

v² = v_o² + 2 a (x - x_o)

Note that x_o and v_o are displacement and velocity at the beginning of the time interval. The symbols x and v represent the displacement and velocity at the end of the time interval. In these equations, t is the time at the end of the interval, or the elapsed time. The value of t_o has been arbitrarily set to zero, as one might do with a stopwatch. The constant acceleration is represented by "a." Thus we note that the above equations are set up for a time interval during which the acceleration must remain constant. Graphs representing this situation appear in Figure 1

Freely Falling Objects

If an object falls near the surface of the earth (where we live) and experiences no force except the force of gravity, we say it is in Free Fall. Since that force is approximately constant near the surface of the earth, we find that its acceleration is a constant with a value of 9.81 m/s², or about 32.2 ft/s². We call this the acceleration due to gravity only. Our text does not use the word "only," but its use is advisable since if there is any other significant force the acceleration is no longer 9.81 m/s².

We give a comment on the use of the word "approximately" above. Consider Boston at an elevation of sea level, and Denver at an elevation of about one mile above sea level. Careful consideration shows that if Denver has a value of g = 9.8000 m/s², then strickly speaking, the value at Boston is g = 9.8005 m/s², or 0.05% higher. Ordinarily, we perform calculations to three digit accuracy (three significant digits); to that approximation the above two numbers are not different. The difference occurs in the fifth digit. Thus we say that the acceleration, g, is approximately a "constant."

Motion with Nonconstant Acceleration

If the acceleration is not constant, the above equations will not hold. We then need a more complicated approach using the techniques of calculus. Integrals must be done. We will not pursue that here.