# 13.2: Describing One-Dimensional Motion

**At Grade**Created by: CK-12

We will start by describing one-dimensional motion, even though the LAS is embedded in a three-dimensional world. As long ago as the 17th century, Galileo Galilei (1564-1642) made great progress in understanding motion in three-dimensions by breaking the problem into separate one-dimensional problems. From centuries of study, scientists have determined that position, velocity, and acceleration are the important and necessary quantities to describe motion. See the Kinematics chapter in this book for more discussion of one-dimensional motion: motion diagrams, observing motion with motion sensors, graphing of motion and understanding graphs of motion. Also see Laboratory Activities in this book for experiments on motion.

## Position and Displacement

Position is where you are in space. It is measured with respect to a coordinate origin using MKS units of meters. When simulating the one-dimensional rocket, we will be interested in the up and down position of the rocket. We often call that the y direction and the position value y. Displacement is defined as the difference in position between two elapsed times. Displacement differs from our concept of distance. If we make a round trip going from Minneapolis to Grand Rapids Minnesota, the displacement is zero while the distance traveled is around 650 kilometers. Displacement, not distance, is the crucial concept in understanding motion.

## Velocity

Velocity is how fast you move through space. It is the rate of change of position with time. Average velocity is defined as the displacement divided by the time elapsed. For large elapsed times, average velocity gives us a very rough idea of how rapidly we moved through space and sometimes not even that. For the round trip described above the average velocity is zero even though we may have been moving at a reasonable average rate during the whole trip. The concept of average velocity becomes most useful when we consider its limit as the elapsed time interval approaches zero. Then we get a measure of the rate of motion at the instant in question. We call the limit of the average velocity as the elapsed time approaches zero the instantaneous velocity. When we speak of motion in the y direction we call this v_{y}. If you know calculus, it is the derivative of position with time dy/dt. Graphically, instantaneous velocity is the slope of the tangent line to the y vs. t curve at the time in question (see the Kinematics chapter for more detail). The concept of instantaneous velocity is essential to a further understanding of motion.

## Acceleration

Velocities aren't always steady - they often change with time. Sometimes the magnitude (or absolute value) of the velocity increases, like when you drop a ball, and sometimes velocities decrease or stay steady. It is valuable to realize that we use the same process in defining rates of change of velocity as we did above when defining rates of change of position. Acceleration is the term we use for rate of change of velocity. Average acceleration is the change in velocity divided by the elapsed time. Again it is instantaneous acceleration, or the limit of the average acceleration as the elapsed time approaches zero, that tells the best story about motion. If you know calculus, it is the derivative of velocity with time dv_{y}/dt. Graphically, instantaneous acceleration is the slope of the tangent line to the v_{y} vs. t curve at the time in question. On Earth, freely falling objects accelerate downward at roughly 10 m/s^{2}. Once the rocket engines cut off, the only force on the rocket is gravity (if we neglect air resistance) and we say the rocket is in freefall accelerating downward at a constant rate – even if its motion continues upward for some time.

## Positive and Negative

It is important to assign a positive or negative sign to the values of position, velocity and acceleration. It makes a big difference whether the rocket is going upward or downward. We will choose upward as positive. You could choose downward as positive as long as you are consistent, but every sign would be interpreted oppositely. For position, picking upward positive means that everything above the origin (launch pad) is positive and everything below is negative. For velocity, moving upward is positive and moving downward is negative. Finally, the acceleration of the rocket during the burn is positive because we are increasing its velocity upward. Once the burn stops, the acceleration is negative because gravity pulls the rocket downward. Use the concepts of positive and negative when you analyze and interpret your data and graphs later.

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