HOW THINGS MOVE-POSITION, VELOCITY, AND ACCELERATION

"To understand motion is to understand nature." -Leonardo da Vinci

One of physics' first problems was extremely practical: at what angle should you launch a cannonball to hit a particular target? Certainly this doesn't sound as grand or sophisticated as analyzing the whole universe, but it does fit nicely with chemistry's roots: attempting to turn base metals into gold. Cannons came into the picture when people went after someone else's gold.

Although physics' early history is rich in characters and their stories, we will focus here on motion from a more modern perspective. Not only does motion form the basis of much of physics' later developments, but it also has a direct connection to human experience. We are surrounded by moving things and experience a great deal of personal motion-even it it's not always in the right direction.

Let's start with the simplest motion first.

Motion Along a Line (one-dimensional motion)

Suppose you live on a road that runs straight east and west. A dog sitting on your road could be located by how far it is from your house. Since physics uses the metric system, the dog might be found, say, 39 meters (almost 130 feet) east of your house. This is called the dog's position. Graphically, the street is represented by a single line, called the x-axis, and your house is located at the origin. East is conventionally regarded as positive and west as negative.

dog position

If a body changes its position, that change is called a displacement. Representing this mathematically, the earlier position is called [x.sub.1]/1, read as x sub 1, while the later position is [x.sub.2], and the displacement is [x.sub.2]- [x.sub.1], represented by [delta]x, read delta x.

To summarize conceptually and mathematically, and including units:

Displacement is the change in an object's position. [delta]x = [x.sub.2] - [x.sub.1]

The meter is a convenient unit for length measurement since the height of most humans is between 1.5 m (5 ft) and 2 m (6 ft 7 in), the length of a normal stride is almost 1 m (3 ft), and a stack of forty books like this one would be 1 m (3 ft) tall.

For example, if the dog moved from its 39 m spot to a position 51 m (170 ft) east of your house, the dog's displacement would be 51 m - 39 m = 12 m.

dog displacement

Next, the elapsed time must be taken into account. The velocity of a body tells how fast a body's position is changing.

Velocity is the distance traveled by a body divided by the time required to travel it.

v = [delta]x/[delta]t units = meters/seconds = m/s

A velocity of 1 m/s (3 ft/s) corresponds to a leisurely stroll (Frankenstein set this standard), while 10 m/s (30 ft/s) is achieved by 100 m (100 yd) dash champions. Your flying disc tosses are usually somewhere between 3 and 10 m/s, as you found in Experiment 1. Speed and velocity are used interchangeably in linear motion, but in two- or three-dimensional motion, there is an important difference-and that is direction.

Velocity is such an integral feature of modern life that automobiles have a device to measure their speed directly, the speedometer. The speedometer's units (miles/hr = mph or km/hr) differ from the scientific standard m/s, but the conversion is quite simple: 60 mph is about 100 km/hr or almost 30 m/s (44 ft/s). Riding in a car at speeds of 0 to 30 m/s seems pretty ordinary, but some bodies travel a lot slower. For example, human hair grows at a few centimeters per month and currents in the Earth's mantle cause earthquakes by dragging crustal plates at a speed of a few centimeters per year. Much faster speeds are also possible: passenger jets fly more than 200 m/s (600 ft/s); the fastest airplane travels at 3,000 m/s (9,000 ft/s); Earth's motion around the sun requires 30,000 m/s (90,000 ft/s); and the ultimate speed is that of light, 300 million m/s (900 million ft/s).

Returning to the example of our friendly dog, if it takes the pooch 4 seconds to travel from 39 m to 51 m, his velocity is 12 m / 4 s = 3 m/s (9 ft/s). This is only an average velocity, since the dog might have run the first 5 meters in 1 second, taken 1 second for a bark break, then run the last 7 meters in 2 seconds.

dog velocity

Next, what about the case where the velocity changes? The rate of change of a body's velocity is called acceleration.

Acceleration is the change in a body's velocity divided by the time required to change it.

a = [delta]v/[delta]t Units = meters/sec/sec = m/[s.sup.2]

Speeding up yields a positive acceleration, while slowing down makes the acceleration negative. Perhaps that's why the gas pedal on your car is called the accelerator. But why isn't the brake called a decelerator? (a mystery we won't solve here).

A value of 1 m/[s.sup.2] is a mild acceleration corresponding to having your car accelerate from 0 to 60 mph in 28 seconds. (That's not much of a jackrabbit start, but it's easy on gas.) A world's record 0 to 60 mph acceleration is approximately 10 m/[s.sup.2], about the same acceleration as a dropped object near the Earth's surface falling because of gravity. That special acceleration of freely falling bodies is called the acceleration due to gravity and given the symbol g. For calculations, g = 9.8 m/[s.sup.2] is used. Humans have built-in limitations on the amount of acceleration we can stand. Blackouts and possible internal damage can occur with accelerations much beyond 60 m/[s.sup.2] (also called 6 g's because it is 6 times the acceleration due to gravity).

If our friendly dog travels at 3 m/s, then is observed 5 seconds later traveling 6 m/s, it accelerated at a rate of (6 m/s - 3 m/s)/5 s = 0.6 m/[s.sup.2].

dog acceleration

Combining the position, velocity, and acceleration in a slightly different way.

An accelerating body's velocity is changed by an amount equal to the acceleration times the elapsed time.

v = [v.sub.0] + at, where [v.sub.0] = velocity at time = 0

The distance traveled is also affected by acceleration:

The position of an accelerated body is increased by its velocity times the elapsed time plus half the acceleration times the elapsed time squared.

x = [x.sub.0] + [v.sub.0]t + 1/2 [at.sup.2]

where [x.sub.0] and [v.sub.0] are the body's position and velocity at time = 0

Another equation may be obtained by eliminating time from the x and v equations:

[v.sub.2] = [v.sub.0.sup.2] + 2a(x - [x.sub.0])

where [x.sub.0] and [v.sub.0] are the body's position and velocity at time = 0

This equation is particularly useful in solving problems in which time is not involved.

Understanding motion along a line is a good start, but it's not enough. Even a dog escapes its straight line occasionally.

Motion in Two or Three Dimensions

Two-dimensional motion in a horizontal plane includes chess pieces moving on a board, a hockey puck slid on the ice, or even a dog running in a flat backyard. The motion's direction has now become important and is sometimes specified in compass cardinal points. Motion in a vertical plane is perhaps more interesting because Earth's gravity pulls everything vertically downward. Examples of this kind of motion encompass not only the classic cannonballs but also movements made in many sports, such as baseball, basketball, football, golf, soccer, and tennis. In principle, analysis of the motion of any of these examples is simple. Dividing the motion into two parts, horizontal (x) and vertical (y), the analytical tools for one-dimensional motion are applied twice, once for each dimension. The resulting path of the ball, called its trajectory, is clearly curved, undoubtedly corresponding to your ball-playing experience.

ball trajectory

Using the equations developed earlier, the range, x, can be found if you know the launch velocity, [v.sub.o], the launch angle, [theta], and the acceleration due to gravity, g. It is:

x = [v.sub.o.sup.2]sin, 2[theta]/g, where sin is the shorthand notation for sine = opposite/hypotenuse

This equation can be solved for the launch angle as a function of the range and launch velocity. Or, knowing the range and angle, you can find the launch velocity, as you saw in Experiment 2. So, could you use this knowledge to become a technical adviser, telling pirates where to aim their cannons, or a caddy for Tiger Woods, advising him on the angle his club should have? Not quite. Other factors are at work besides gravity. In reality, interactions with air make all shots fall short of their expected range or even curve right or left, turning their motion into three-dimensional rather than two-dimensional movement. So, what causes bodies to move? The short answer is: forces. The longer answer is contained in the next chapter.

Experiment 3: Weight 'Til the Sun Shines, Nellie

Equipment needed: spring scale calibrated in newtons and grams and several objects (see table)

The purpose of this experiment is to measure the weight and mass of several household objects so you can become familiar with standard physics units.

Procedure

Hang an object from a spring scale calibrated in newtons (weight) and grams (mass) and record your values in the table below:

Object Weight Mass Mass = (newtons) (grams) grams/1,000 (kilograms)