Modeling One- Dimensional Motion W o r d s P i c t u r e s Graphs D a t a t a b l e s E q u a t i o n s

Verbal Model Common words and phrases used to describe the motion of an object in a straight line (one- dimensional motion): speeding up; slowing down constant speed; constant velocity accelerating; decelerating distance; displacement speed; velocity The proper use of these terms in a scientific sentence may be quite different from their use in everyday conversation.

Verbal Model – Constant Speed The position of any object must be given with respect to some reference point. An object’s position is its directed distance from a reference point. Movement is said to have occurred when the position of an object with respect to a given reference point has changed.

Displacement, then, is a vector quantity. Distance, or length, is a scalar. These two terms, distance and displacement, do not mean the same thing! This change in position of an object is often called its displacement. It is the straight line distance and direction from the starting point to the ending point, regardless of the path taken.

In one dimensional motion, the displacement direction is often given as positive, +, or negative, -. A displacement of +3.5 m implies movement of 3.5 m in the positive direction. A displacement of -3.5 m implies movement of 3.5 m in the negative direction. Positive and negative directions are chosen arbitrarily, but usually agree with standard mathematical conventions. standard mathematical conventions.

While some people think that the word “velocity” is just a fancy word for “speed,” the two words have different meanings in science. Speed simply tells you how fast the object is moving. Velocity tells how fast and in what direction. If you tell someone that a car is traveling 50 mph, you have specified a speed. If you say 50 mph south, you have specified a velocity.

Average speed is the total distance traveled divided by the total time taken. Average velocity is the total displacement divided by the total time taken. Average speed and average velocity are generally not equivalent because total distance and total displacement are generally not the same. When would they have the same magnitudes?

The average velocity of an object is defined to be the ratio of its change in position (displacement) to the time taken to change the position. v av v av = average velocity; in units of m/s, mph, ft/s, km/hr, etc... in units of m/s, mph, ft/s, km/hr, etc... = d d = change in position, or displacement; in units of m, in, ft, km, mi, etc... in units of m, in, ft, km, mi, etc... t t = change in time; in units of s, min, hr, etc... in units of s, min, hr, etc...

positive velocity A positive velocity indicates movement positive direction in the positive direction. negative velocity A negative velocity indicates movement negative direction in the negative direction. The “sign” of the velocity indicates the direction of movement.

Pictorial Model – Constant Speed One of the simplest ways to give a pictorial model of one-dimensional motion is to draw a series of dots. Each successive dot is assumed to occur after identical amounts of time. The spacing of the dots is proportional to the distance the object traveled between each time period. Therefore, if the dots are evenly spaced, the object represented by the dots was moving at a constant speed. The greater the spacing of the dots, the faster the object was traveling.

Numerical Model – Constant Speed Imagine that a pair of numbers are assigned to each dot in the pictorial model. One number designates time, and the other number designates the position of the object at that time. Since the dots are evenly spaced, the numbers assigned to those dots will also be evenly spaced. You now have a numerical model (data table) representing the object’s motion.

Graphical Model – Constant Speed When the ordered pairs associated with the pictorial model are graphed on a coordinate plane in which the position is the y-coordinate and the time is the x-coordinate, the graph will be linear. A linear “Position vs Time” graph therefore models an object moving with a constant speed. The “steepness” of the graph is proportional to the object’s speed.

Mathematical Model – Constant Speed The equation of the line modeling the object with constant speed will have the general form y = mx + b. The slope of this line (m) is the velocity of the object. The y-intercept (b) is the initial position of the object. Since y is the position and x is the time, our equation becomes x(t) = vt + x i, where x(t) is the position at time t, v is the velocity, and x i is the initial position.