One Dimensional Motion

Physics concerns intself with a variety of broad topics. One such topic is mechanics - the study of the motion of objects
Kinematics is the science of describing the motion of objects using words, diagrams, numbers, graphs, and equations. Kinematics is a branch of mechanics. The goal of any study of kinematics is to develop sophisticated mental models which serve to describe (and ultimately, explain) the motion of real-world objects.
Scalars and Vectors
Scalars are quantities which are fully described by a magnitude (or numerical value) alone.
Vectors are quantities which are fully described by both a magnitude and a direction.
Distance and Displacement
Distance and displacement are two quantities which may seem to mean the same thing yet have distinctly different definitions and meanings.
Distance is a scalar quantity which refers to "how much ground an object has covered" during its motion.
Displacement is a vector quantity which refers to "how far out of place an object is"; it is the object's overall change in position.
Speed and Velocity
Speed is a scalar quantity which refers to "how' fast an object is moving." Speed can be thought of as the rate at which an object covers distance.
Velocity is a vector quantity which refers to "the rate at which an object changes its position." velocity is direction aware. When evaluating the velocity of an object, one must keep track of direction.
Calculating Average Speed and Average Velocity
The average speed during the course of a motion is often computed using the following formula:
Average Speed = Distance Traveled / Time of Travel
Meanwhile, the average velocity is often computed using the equation
Average Velocity = position / time = displacement / time
Average Speed versus Instantaneous Speed
Since a moving object often changes its speed during its motion, it is common to distinguish between the avel'age speed and the instantaneous speed. The distinction is as follows.
Instantaneous Speed - the speed at any given instant in time.
Average Speed - the average of all instantaneous speeds; found simply by a distance / time ratio.
one 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
Acceleration is a vector quantity which is defined as the rate at which an object changes its velocity. An object is accelerating ifit is changing its velocity.
The Meaning of Constant Acceleration
Sometimes an accelerating object will change its velocity by the same amount each second .. This is referred to as a constant acceleration since the velocity is changing by a constant amount each second.
Calculating the Average Acceleration
The average acceleration (a) of any object over a given interval of time (t) can be calculated using the equation
Ave. acceleration = velocity / time = vf - vi / t
Acceleration values are expressed in units of velocity/time. Typical acceleration units include the following:
m/s/s
mi/hr/s
km/hr/s
m/s2
The Direction of the Acceleration Vector
Since acceleration is a vector quantity, it has 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
The shapes afthe pasition versus time graphs for these two.basic types af matian - canstant velacity matian and accelerated mati an (i.e., changing velacity) - reveal an impartant principle. The principle is that the slape afthe line an a pasitian-time graph reveals useful infarmatian abaut the velacity af the abject. It is aften said, "As the slope goes, so. gaes the velacity." Whatever characteristics the velacity has, the slape will exhibit the same (and vice versa). If the velacity is canstant, then the slape is canstant (i.e., a straight line). If the velacity is changing, then the slape is changing (i.e., a curved line). If the velacity is pasitive, then the slape is pasitive (i.e., maving upwards and to. the right). This very principle can be extended to.any matian conceivable.
The Meaning of Slope for a p-t Graph
The slape af a pasitian vs. time graph reveals pertinent infarmatian abaut an abject's velacity. For example, a small slape means a small velacity; a negative slape means a negative velacity; a canstant slape (straight line) means a canstant velacity; a changing slape (curved line) means a changing velacity. Thus the shape af the line an the graph (straight, curving, steeply slaped, mildly slaped, etc.) is descriptive afthe abject's matian.
Determining the Slope on a p-t Graph
Let's begin by cansidering the pasitian versus time graph belaw.
The line is slaping upwards to. the right. But mathematically, by haw much daes it slape upwards far every 1 secand alang the horizantal (time) axis? To. answer this questian we must use the slape equatian.
Slope = y/x = y2-y1/x2-x1 = rise/run
The slope equation says that the slope of a line is found by determining the amount of rise of the line between any two points divided by the amount of run of the line between the same two points. In other words,
pick two points on the line and determine their coordinates.
Determine the diffreence in y-coordinates of these two points (rise).
Determine the difference in x-coordinates for these two points (run).
Divide the diffreence in y-coordinates by the difference in x-coordinates (rise/run or slope).
The Meaning of Shape for a v-t Graph
The specific features of the motion of objects are demonstrated by the shape and the slope of the lines on a velocity vs. time graph.
Consider a car moving with a constant, rightward (+) velocity - say of + 10 mls. As learned in an earlier, a car moving with a constant velocity is a car with zero acceleration
If the velocity-time data for such a 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 zero slope (a horizontal line has zero slope) when plotted as a velocity-time graph. Furthermore, only positive velocity values are plotted, corresponding to a motion with positive velocity.
Now consider a car moving with a rightward (+), changing velocity - that is, a car that is moving rightward but speeding up or accelerating. Since the car is moving in the positive direction and speeding up, the car is said to have a positive acceleration.
If the velocity-time data for such a 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 sloped line when plotted as a velocity-time graph. The slope of the line is posItive, corresponding to the positive acceleration. Furthermore, only positIve velocity values are plotted, corresponding to a motion with positive velocity.
The velocity vs. time graphs for the two types of motion - constant velocity and changing velocity (acceleration) - can be summarized as follows.
The Importance of Slope
The shapes of the velocity 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 velocity-time graph reveals useful information about the acceleration of the object. If the acceleration is zero, then the slope is zero (i.e., a horizontal line). If the acceleration is positive, then the slope is positive (i.e., an upward sloping line). If the acceleration is negative, then the slope is negative (i.e., a downward sloping line). This very principle can be extended to any conceivable motion.
The slope of a velocity-time graph reveals information about an object's acceleration. But how can one tell whether the object is moving in the positive direction (i.e., positive velocity) or in the negative direction (i.e., negative velocity)? And how can one tell if the object is speeding up or slowing down?
The answers to these questions hinge on one's ability to read a graph. Since the graph is a velocity-time graph, the velocity would be positive whenever the line lies in the positive region (above the x-axis) of the graph. Similarly, the velocity would be negative whenever the line lies in the negative region (below the x-axis) of the graph., a positive velocity means the object is moving in the positive direction; and a negative velocity means the object is moving in the negative direction. So one knows an object is moving in the positive direction if the line is located in the positive region of the graph (whether it is sloping up or sloping down). And one knows that an object is moving in the negative direction if the line is located in the negative region of the graph (whether it is sloping up or sloping down). And finally, if a line crosses over the x-axis from the positive region to the negative region of the graph (or vice verse), then the object has changed directions.
Now how can one tell if the object is speeding up or slowing down? Speeding up means that the magnitude (or numerical value) of the velocity is getting large. For instance, an object with a velocity changing from +3 m/s to + 9 m/s is speeding up. Similarly, an object with a velocity changing from -3 m/s to -9 m/s is also speeding up. In each case, the magnitude of the velocity (the number itself, not the sign or direction) is increasing; the speed is getting bigger. Given this fact, one would believe that an object is speeding up if the line on a velocity-time graph is changing from near the 0-velocity point to a location further away from the 0-velocity point. That is, if the line is getting further away from the x-axis (the 0-velocity point), then the object is speeding up. And conversely, if the line is approaching the x-axis, then the object is slowing down.
Determining the Slope on a v-t Graph
The slope of the line on a velocity versus time graph is equal to the acceleration of the object. If the object is moving with an acceleration of +4 m/s/s (i.e., changing its velocity by 4 m/s per second), then the slope of the line will be +4 m/s/s. If the object is moving with an acceleration of -8 mls/s, then the slope of the line will be -8 mls/s. If the object has a velocity of 0 m/s, then the slope of the line will be 0 m/s. Because of its importance, a student of physics must have a good understanding of how to calculate the slope of a line. In this part of the lesson, the method for determining the slope of a line on a velocity-time graph will be discussed.
Let's begin by considering the velocity ver-sustime graph below.
The line is sloping upwards to the right. But mathematically, by how much does it slope upwards for every 1 second along the horizontal (time) axis? To answer this question we must use the slope equation.
Slope = y/x = y2-y1/x2-x1 = rise/run
The slope equation says that the slope of a line is found by determining the amount of rise of the line between any two points divided by the amount of run of the line between the same two points. A method for carrying out the calculation is
Pick two points on the line and determine their coordinates.
Determine the difference in y-coordinates for these two points (rise).
Determine the difference in x-coordinates for these two points (run).
Divide the difference in y-coordinates by the difference in x-coordinates (rise/run or slope).
Determining the Area on a v-t Graph
As learned in an earlier part, a plot of velocity-time can be used to determine the acceleration of an object (the slope). In this part, we will learn how a plot of velocity versus time can also be used to determine the displacement of an object. For velocity versus time graphs, the area bound by the line and the axes represents the displacement. The diagram below shows three different velocity-time graphs; the shaded regions between the line and the time-axis represents the displacement during the stated time interval.
The shaded area is representative of the displacement during form 0 seconds to 6 seconds. This area takes on the shape of a rectangle can be calculated using the appropriate equation.
The shaded area is representative of the displacement during from 0 seconds to 4 seconds. This area takes on the shape of a triangle can be calculated using the appropriate equation.
The shaded area is representative of the displacement during from 2 seconds to 5 seconds. This area takes on the shape of a trapezoid can be calculated using the appropriate equation.
The method used to find the area under a line on a velocity-time graph depends upon whether the section bound by the line and the axes is a rectangle, a triangle or a trapezoid. Area formulas for each shape are given below.
Introduction to Free Fall
A free-falling object is an object which is falling under the sole influence of gravity. Any object which is being acted upon only be the force of gravity is said to be in a state of free fall. There are two important motion characteristics which are true of free-falling objects:
Free-falling objects do not encounter air resistance.
All free-falling objects (on Earth) accelerate downwards at a rate of 9.8 m/s/s (often approximated as 10 m/s/s for back-of-the-envelope calculations)
The Acceleration of Gravity
It was learned in the previous part of this lesson that a free-falling object is an object which is falling under the sole influence of gravity. A free-falling object has an acceleration of 9.8 mls/s, downward (on Earth). This numerical value for the acceleration of a free-falling object is such an important value that it is given a special name. It is known as the acceleration of gravity - the acceleration for any object moving under the sole influence of gravity. A matter of fact, this quantity known as the acceleration of gravity is such an important quantity that physicists have a special symbol to denote it - the symbol g. The numerical value for the acceleration of gravity is most accurately known as 9.8 m/s/s. There are slight variations in this numerical value (to the second decimal place) which are dependent primarily upon on altitude. We occasionally use the approximated value of 10 m/s/s in The Physics Classroom Tutorial in order to reduce the complexity of the many mathematical tasks
g = 9.8 m/s/s, downward
( ~ 10 m/s/s, downward)
The Big Misconception
we have learnt that the acceleration of a free-falling object (on earth) is 9.8 m/s/s. This value (known as the acceleration of gravity) is the same for all free-falling objects regardless of how long they have been falling, or whether they were initially dropped form rest or throuwn up into the air. Yet the questions are often asked "doesn't a more massive object accelearate at a greater rate than a less massive object?" " Wouldn't an elephant free-fall faster than a mouse?" This question is a reasonable inquiry that is probable based in part upon personal observations made of falling objects in the physical world. After all, nearly everyone has observed the didderence in the rate of fall of a single piece of paper (or similar object) and a textbook. The two objects clearly travel to the ground at different rates - with the more massive book falling faster.
The answer to the question (doesn't a more massive object accelerate at a greater rate than a less massive object?) is absolutely not! That is, absolutely not if we are considering the specific type of falling motion known as free-fall. Free-fall is the motion 6f objects which move under the sole influence of gravity; free-falling objects do not encounter air resistance. More massive objects will only fall faster if there is an appreciable amount of air resistance present.
The actual explanation of why all objects accelerate at the same rate involves the concepts of force and mass. The details will be discussed later. At that time, you will learn that the acceleration of an object is directly proportional to force and inversely proportional to mass. Increasing force tends to increase acceleration while increasing mass tends to decrease acceleration. Thus, the greater force on more massive objects is offset by the inverse influence of greater mass. Subsequently, all objects free fall at the same rate of acceleration, regardless of their mass
The Kinematic Equations
d = vi*t + 1/2*a*t2 vf2 = vi2 + 2*a*d
vf = vi + a*t d = vi + vf/2 * t
There are a variety of symbols used in the above equations. Each symbol has its own specific meaning. The symbol d stands for the displacement of the object. The symbol t stands for the time for which the object moved. The symbol a stands for the acceleration of the object. And the symbol v stands for the velocity of the object; a subscript of i after the v (as in vi) indicated that the velocity value is the initial velocity value and a subscript of f (as in vf) indicates that the velocity value is the final velocity value.
Each of these four equations appropriately describe the mathematical relationship between the parameters of an object's motion. As such, they can be used to predict unknown information about an object's motion if other information is known.
Kinematic Equations and Free Fall
As mentioned, a free-falling object is an object which is falling under the sale influence of gravity. That is to say that any object which is moving and being acted upon only be the force of gravity is said to be "in a state of free fall." Such an object will experience a downward acceleration of 9.8 m/s/s. Whether the object is falling downward or rising upward towards its peak, if it is under the sole influence of gravity, then its acceleration value is 9.8 m/s/s.
Like any moving object, the motion of an object in free fall can be described by four kinematic equations. The kinematic equations which describe any object's motion are:
The Kinematic Equations
d = vi*t + 1/2*a*t2 vf2 = vi2 + 2*a*d
vf = vi + a*t d = vi + vf/2 * t
The symbols in the above equation have a specific meaning: the symbol d stands for the displacement; the symbol t stands for the time; the symbol a stands for the acceleration of the object; the symbol vi stands for the intitial velocity value; and the symbol vf stands for the final velocity.
There are a few conceptual characteristics of free fall motion which will be of value when using the equations to analyze free fall motion. These concepts are described as follows:
An object in free fall experiences an acceleration of -9.8 mis/so (The sign indicates a downward acceleration.) Whether explicitly stated or not, the value of the aceeleration in the kinematic equations is -9.8 rnIs/s for any freely falling object.
If an object is merely dropped (as opposed to being thrown) from an elevated height, then the initial velocity of the object is 0 m/s.
If an object is projected upwards in a perfectly vertical direction, then it will slow down as it rises upward. The instant at which it reaches the peak of its trajectory, its velocity is 0 m/s. This value can be used as one of the motion parameters in the kinematic equations; for example, the final velocity (Vf) after traveling to the peak would be assigned a value of 0 m/s.
If an object is projected upwards in a perfectly vertical direction, then the velocity at which it is projected is equal in magnitude and opposite in sign to the velocity which it has when it returns to the same height. These four principles and the four kinematic equations can be combined to solve problems involving the motion of free falling objects.