Notes: Linear Motion

Objective:
To gain an understanding of the concepts of straight line motion.

Notes:
Vectors and Scalars
A study of motion will involve the introduction of a variety of quantities which are used to describe the physical world. Examples of such quantities include distance, displacement, speed, velocity, acceleration, force, mass, momentum, energy, work, power, etc. All these quantities can by divided into two categories vectors and scalars. A vector quantity is a quantity which is fully described by both magnitude and direction. On the other hand, a scalar quantity is a quantity which is fully described by its magnitude. The emphasis of this unit is to understand some fundamentals about vectors and to apply the fundamentals in order to understand motion and forces which occur in two dimensions.

• Scalars are quantities which are fully described by a magnitude alone.
• Vectors are quantities which are fully described by both a magnitude and a direction.

Test your understanding of this distinction, between vectors and scalars. Consider the following quantities listed below. Depress your mouse on the pop-up menu next to each quantity to view the answer.
 

Speed 
Speed is a scalar quantity which refers to "how fast an object is moving." 

Velocity
Velocity is a vector quantity which refers to "the rate at which an object changes its position." Imagine a person moving rapidly - one step forward and one step back - always returning to the original starting position. While this might result in a frenzy of activity, it would result in a zero velocity.

Velocity is a vector quantity. As such, velocity is "direction-aware." When evaluating the velocity of an object, one must keep track of direction. 

It would not be enough to say that an object has a velocity of 55 mi/hr. One must include direction information in order to fully describe the velocity of the object. For instance, you must describe an object's velocity as being 55 mi/hr, east. 

Speed verses Velocity verses Acceleration
This is one of the essential differences between speed and velocity. Speed is a scalar and does not keep track of direction; velocity is a vector and is direction-aware.

The task of describing the direction of the velocity vector is easy. The direction of the velocity vector is simply the same as the direction which an object is moving. It would not matter whether the object is speeding up or slowing down, if the object is moving rightwards, then its velocity is described as being rightwards. If an object is moving downwards, then its velocity is described as being downwards.

So an airplane moving towards the west with a speed of 300 mi/hr has a velocity of 300 mi/hr, west.

Note that speed has no direction (it is a scalar) and velocity is simply the speed with a direction.

The average speed during the course of a motion is often computed using the following equation:

Try the following problem:

To compute her average speed, we simply divide the distance of travel by the time of travel.

v = distance divided by time = 200/4 = 50 miles per hour

Elvis averaged a speed of 50 miles per hour. He may not have been traveling at a constant speed of 55 mi/hr. He undoubtedly, was stopped at some instant in time (perhaps for a bathroom break or for lunch) and he probably was going 65 mi/hr at other instants in time. Yet, he averaged a speed of 50 miles per hour.

Since a moving object often changes its speed during its motion, it is common to distinguish between the average speed and the instantaneous speed. The distinction is as follows.

Instantaneous Speed - speed at any given instant in time.

Average Speed - average of all instantaneous speeds; found simply by a distance/time ratio.

You might think of the instantaneous speed as the speed which the speedometer reads at any given instant in time and the average speed as the average of all the speedometer readings during the course of the trip.

Acceleration is a vector quantity which is defined as "the rate at which an object changes its velocity." An object is accelerating if it is changing its velocity.

Sports announcers will occasionally say that a person is accelerating if he/she is moving fast. Yet acceleration has nothing to do with going fast. 

A person can be moving very fast, and still not be accelerating. Acceleration has to do with changing how fast an object is moving. If an object is not changing its velocity, then the object is not accelerating. 

The acceleration of any object is calculated using the equation

Acceleration = Change in velocity divided by time

Acceleration values are expressed in units of velocity/time. Typical acceleration units include the following:

meters/second square = m/s²

Since acceleration is a velocity change over a time, the units on acceleration are velocity units divided by time units - thus (m/s)/s or (mi/hr)/s.

Since acceleration is a vector quantity, it will always have a direction associated with it. The direction of the acceleration vector depends on two things:

• whether the object is speeding up or slowing down
• whether the object is moving in the + or - direction

 p-t Graphs

These notes focus on the use of position vs. time graphs to describe motion. The specific features of the motion of objects are demonstrated by the shape and the slope of the lines on a position vs. time graph. The first part of this lesson involves a study of the relationship between the shape of a p-t graph and the motion of the object.

To begin, consider a car moving with a constant, rightward (+) velocity - say of +10 m/s.

If the position-time data for this car were graphed, then the resulting graph would look like the graph at the right. Note that a motion described as a constant, positive velocity results in a line of constant and positive slope when plotted as a position-time graph.

Now consider a car moving with a rightward (+), changing velocity - that is, a car that is moving rightward but speeding up or accelerating.

If the position-time data for this car were graphed, then the resulting graph would look like the graph at the right. Note that a motion described as a changing, positive velocity results in a line of changing and positive slope when plotted as a position-time graph.

The position vs. time graphs for the two types of motion - constant velocity and changing velocity (acceleration) - are shown as follows.

Constant Velocity Positive Velocity	Positive Velocity Changing Velocity (acceleration)

 

The shapes of the position vs. time graphs for these two basic types of motion - constant velocity motion and accelerated motion (i.e., changing velocity) - reveal an important principle. The principle is that the slope of the line on a position-time graph reveals useful information about the velocity of the object. It's often said, "As the slope goes, so goes the velocity." Whatever characteristics the velocity has, the slope will exhibit the same (and vice versa). If the velocity is constant, then the slope is constant (i.e., a straight line). If the velocity is changing, then the slope is changing (i.e., a curved line). If the velocity is positive, then the slope is positive (i.e., moving upwards and to the right). This very principle can be extended to any motion conceivable.

Consider the graphs below as example applications of this principle concerning the slope of the line on a position vs. time graph. The graph on the left is representative of an object which is moving with a positive velocity (as denoted by the positive slope), a constant velocity (as denoted by the constant slope) and a small velocity (as denoted by the small slope). The graph on the right has similar features - there is a constant, positive velocity (as denoted by the constant, positive slope). However, the slope of the graph on the right is larger than that on the left; this larger slope is indicative of a larger velocity. The object represented by the graph on the right is traveling faster than the object represented by the graph on the left. The principle of slope can be used to extract relevant motion characteristics from a position vs. time graph; as the slope goes, so goes the velocity.

 

Slow, Rightward (+) Constant Velocity	Fast, Rightward (+) Constant Velocity

Consider the graphs below as another application of this principle of slope. The graph on the left is representative of an object which is moving with a negative velocity (as denoted by the negative slope), a constant velocity (as denoted by the constant slope) and a small velocity (as denoted by the small slope). The graph on the right has similar features - there is a constant, negative velocity (as denoted by the constant, negative slope). However, the slope of the graph on the right is larger than that on the left. Once more, this larger slope is indicative of a larger velocity. The object represented by the graph on the right is traveling faster than the object represented by the graph on the left.

 

Slow, Leftward (-) Constant Velocity	Fast, Leftward (-) Constant Velocity

As a final application of this principle of slope, consider the two graphs below. Both graphs show plotted points forming a curved line. Curved lines have changing slope; they may start with a very small slope and begin curving sharply (either upwards or downwards) towards a large slope. In either case, the curved line of changing slope is a sign of accelerated motion (i.e., changing velocity). Applying the principle of slope to the graph on the left, one would conclude that the object depicted by the graph is moving with a negative velocity (since the slope is negative ). Furthermore, the object is starting with a small velocity (the slope starts out with a small slope) and finishes with a large velocity (the slope becomes large). That would mean that this object is moving in the negative direction and speeding up (the small velocity turns into a larger velocity). This is an example of negative acceleration - moving in the negative direction and speeding up. The graph on the right also depicts an object with negative velocity (since there is a negative slope). The object begins with a high velocity (the slope is initially large) and finishes with a small velocity (since the slope becomes smaller). So this object is moving in the negative direction and slowing down. This is an example of positive acceleration.

 

Rightward (+) Velocity; Slow to Fast	Leftward (-) Velocity Fast to Slow

The principle of slope is an incredibly useful principle for extracting relevant information about the motion of objects as described by their position vs. time graph. Once you've practiced the principle a few times, it becomes a very natural means of analyzing position-time graphs.

Summary:
Speed, velocity, and acceleration are all related quantities. These quantities are used to describe motion. This motion can be in a straight line, curved or an object in free fall.

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