Physics 130
West Chester University

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Lesson 3
Describing Motion - Velocity

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Studying Motion

Sample Movie (Quicktime 6 needed) - Wushu (pronounced "woo-shoo") is what the Chinese call their martial arts. Wushu provides an excellent example of many types of motion. The link here contains a video, which while a bit cheesy, shows an example of one of the top Chinese competitors from the 1990's demonstrating a compulsory competition form. As you watch, think of all the different ways in which to describe motion. The physics subdiscipline of kinematics uses precise language in combination with numbers to try to describe motion.

Hint - In physics, we use very precise wording to say what we mean. Often the words that are chosen have very different meanings in society at large. Or, more often, the words meanings are blurred. As an example, consider the words velocity and speed. These are often taken as synonyms. But in reality, speed is how fast you are going in any direction, whereas velocity defines both the speed and the direction of motion.

Vectors represent quantities that have both a magnitude and direction. Scalars represent pure quantities.

Speed is the amount of distance covered in a given time.

S = {|\Delta x| \over \Delta t}

Since speed is a scalar, speed should always be given as a positive number. Speed is the magnitude of the vector velocity. Velocity is represented as a speed and a direction. Thus, we shall write velocity either as:

v_x = {{\Delta x} \over {\Delta t}}

for velocities in the x direction, or:

v_y = {{\Delta y} \over {\Delta t}}

for velocities in the y direction. While, in general, one can choose a coordinate system in any manner one chooses, we shall use the convention that horizontal motions to the right are positive x velocities (vx), and motions to the left are negative x velocities. Likewise, upward motions shall be considered to be positive y velocities (vy) and downward motions are negative y velocities.

Example Given that the change in position \Delta x = 25 m, and the change in time \Delta t = 5 s the velocity is:

v_x = {{25 m} \over {5 s}} = 5 m/s

Notice that the x velocity is positive and thus motion is to the right along the x-axis by our convention.

Example Given that the change in position \Delta x = -25 m, and the change in time \Delta t = 5 s the velocity is:

v_x = {{-25 m} \over {5 s}} = -5 m/s

Notice that now the x velocity is negative and thus motion is to the left along the x-axis by our convention.

In both cases, the speed is 5 m/s (positive).

We can get more use out of this equation. What if you know your initial position (5 m), and you know that you will travel at 5 m/s for 5 seconds? It turns out that this equation can be rearranged to help you determine the place you end up or your final position.

v_x \Delta t = \Delta x = x_f - x_i

Adding the initial position to both sides of this equation, we get:
x_f = x_i + v_x \Delta t

Example
Let's plug in our numbers to determine what our final position will be under these conditions:
x_f = 5 m + (5 m/s) \times (5 s)

x_f = 5 m + (25 {{m \times s} \over s})

(Notice how in the second term, the unit seconds will cancel.)
x_f = 5 m + 25 m = 30 m

Example
Let's change the numbers. What if we are told that we travel at -5 m/s, and that all the other numbers are the same. Instead of traveling to the right, we are now traveling to the left. We would expect our result, xf to reflect this:
x_f = 5 m + (-5 m/s) \times (5 x)

x_f = 5 m + (-25 m) = -20 m

This result, xf = -20 m, is clearly to the left of our previous result, xf = 30 m, as measured on the x-axis.

Average Velocity is the average speed in a given direction. We will sometimes write it as
\bar v = {{x_f - x_i} \over {t_f - t_i}}

where xf is the ending point, xi is the starting point, and tf and ti are the times corresponding to these respective points. Thus in this convention, speed will always be positive, whereas velocity can be negative depending on the starting and ending distances. Velocity is a vector.

Instantaneous speed is speed over a very small time-span. Thus one could go from point A to point B with one step at a time, stopping after each step. There are times at which the speed would be zero, and other times when it would not be zero. However, when looking at the big picture, one could time the motion from point A to point B and also measure the distance. The distance divided by the total measured time would be the average speed. If a direction were included, then the quantity would become the average velocity.

Acceleration
Acceleration is the change in speed over a given increment of time.

a = {{\Delta v} \over {\Delta t}}

This formula represents the average acceleration of an object. As with velocity, if one looks at the acceleration over a very small time period, one is looking at instantaneous acceleration. Acceleration is a vector - meaning it has both magnitude and direction.