Today, we are going to continue to talk about how physicists describe motion. As we've already discussed, this material is a "preliminary" to our discussion of sound. Let me try to indicate why this preliminary is important. As you probably already know, sound is a wave. Sound waves are not visible, unlike water waves that we can see. We will relate our direct experiences with water waves to sound; this is a standard trick in physics: we frequently apply our knowledge about one phenomenon to another. Our direct experiences with water waves tell us that waves can move. This means it's important to quantitatively define what motion entails. Our discussion will include

- Defining instantaneous velocity
- Velocity versus time graphs
- Examples with constant velocity
- The sonic ranger
- Acceleration
- Example with constant acceleration

In our last class meeting we talked about the definition of *average velocity*. In a similar way that we deduced a smooth trajectory by drawing a line through measured positions, we can also come up with a quantity called the *instantaneous velocity* to describe the motion of an object at each instant in time. This quantity is defined graphically. The graph shows the position of an oscillating object versus time. For this example, I choose an oscillating object because oscillations are of fundamental importance to the science of sound. The techniques discussed below are very general, and can be applied to any kind of motion. For future reference, let's say positive x points to the east and negative x points to the west. With the graph of position versus time, we'll define the velocity.

In words, we define the *velocity*

- draw a straight line
*tangent*to the position versus time curve at an instant in time. By tangent, we mean that the line just touches the curve at one point. - the tangent line can be thought of as a hill. The steeper it is, the larger is the velocity. The
*steepness*is measured by the ratio of "rise"/"run". For our graph, the "rise" of the tangent line is a change in position and the "run" of the tangent line is the change in time. - if the tangent line hill points upward (towards increasing x), then the velocity is positive. If the tangent line "hill" points downward (towards smaller x), then the velocity is negative.

Since the *velocity* is defined as the ratio of change in position (with SI units of meters) to the time interval (with SI units of seconds) in which the change occurs, the SI units of velocity are m/s.

We can choose as many different times as we would like on a position versus time graph to determine the velocity of an object. For each time, we repeat the above procedure. In practice, we generally just use the above procedure for a few times and then draw a smooth curve to get a *velocity versus time* graph. A graph of velocity versus time for the object whose position versus time graph is shown above is

We "read" the above graph in the following way

- In the time interval 0<t<0.7 s, the object has a
*positive*velocity, meaning it's moving towards the east (positive x). Note, that the speed of the object (how many cm/s it is going) is getting smaller in the time interval. - At the times 0.7 s and 2.2 s, the object is at rest! This must be the case because the direction of motion is changing at times before and after these points.
- At times 0, 1.5 and 3.0 s, the object has its greatest speed of >20 cm/s.
- In the time interval 0.7<t<2.2 s, the object has a
*negative*velocity, meaning it's moving towards the west (negative x).

We will be talking in much more detail about oscillatory motion next week. It's very important to understand in detail, before we begin discussing wave motion.

Motion with constant velocity is easy to analyze. Since

we can determine one of the three quantities from the other two using a little algebra. For example, how long does it take for a car to go 50 km at a constant velocity of 25 km/hr?

In class, we will be using a device that will allow measurements of position and time of an object. We'll then use a computer to graph that information and also determine the velocity of the object versus time. The *sonic ranger* is a device that generates *ultrasound* waves. Ultrasound waves are at too high a frequency to be audible. Probably the most familiar application of ultrasound devices is in obstetrics. Ultrasound is used for *non-intrusive* imaging of a baby inside the mother's womb and is a powerful diagnostic to monitor the fetus without causing it or the mother harm. In our application, ultrasound waves are emitted at fixed times and reflect off of a moving or stationary object. The reflected waves are detected at some time after they are emitted. By knowing the speed of sound in air, the round trip distance between the ultrasound emitter and the object can be determined at an instant in time. The position and time information is then interfaced to a computer for analysis.

The concept of *acceleration* is familiar to everyone who drives. Stepping on the car's "accelerator pedal" causes it to speed up. It turns out that with the *physics definition* of acceleration, the car's "brake pedal" also causes acceleration, just with the opposite sign: i.e, the car slows down.

Simply put, acceleration is defined as the ratio of *change in velocity* to the *time interval* over which the change occurs. It's important to not mix up velocity and acceleration! They are different physical quantities with different units. Since acceleration is ratio of the change in velocity (with SI units of m/s) to the time interval (with SI units of seconds) in which the change occurs, the SI units of acceleration are m/s^{2}. Just like with our definition of velocity itself, the best way to define acceleration is graphically.

In words, we define the *acceleration*

- draw a straight line
*tangent*to the velocity versus time curve at an instant in time. By tangent, we mean that the line just touches the curve at one point. - the tangent line can be thought of as a hill. The steeper it is, the larger is the acceleration. The
*steepness*is measured by the ratio of "rise"/"run". For our graph, the "rise" of the tangent line is a change in velocity and the "run" of the tangent line is the change in time. - if the tangent line hill points upward (towards increasing v), then the acceleration is positive. If the tangent line "hill" points downward (towards smaller v), then the acceleration is negative.

There are many examples of motion with constant acceleration, the best known being freely falling objects. All objects, independent of their mass or composition have a constant acceleration of g=9.8 m/s^{2} towards the earth. Since

we can determine one of the three quantities from the other two using a little algebra. For example, how long does it take for a car to come to rest from an initial speed of 50 km/hr if it accelerates at -3 m/s^{2}?

Our first job is to convert the initial speed into SI units. From there, we can use our formula to calculate the time.

Next, we have to find the *change* in velocity and divide it by the acceleration to find the time interval

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Last updated: `4 Sep 1999
`Comments: bland@indiana.edu