M1. Motion along a straight line
This is the simplest type of motion studied in Physics. The trajectory of this motion is a straight line.
Trajectory: the path traced by a moving object.
To describe the motion we need to know the distance traveled by the moving element and time spent to move along this distance.
Displacement: change of the position of a moving object from one position x1 to another position x2.
Displacement has magnitude and direction. Its magnitude is a distance between starting x1, and finishing x2 positions. Its direction is along the straight line drawn from x1 to x2.
In case of motion along the straight line direction is defined by a sign: “+” means displacement in the positive direction, “-“ means displacement in the negative direction. We are talking about the direction of the axis used to describe the motion.
For a mathematical description the moving element is chosen to be a point, which can be represented by a dot on a graph. If the moving object is a physical one – a car, falling stone, walking person – we must consider all points of the object moving in the same direction with the same speed. Then all descriptions of the motion of such objects are made for one arbitrary chosen point of this object. Usually it will be point known as the center of mass and its position is well defined for any object of any type.
The simplest case – motion with constant velocity.
Let’s consider motion of an object along a horizontal x axis, pointing from left to right. Using x1 and x2 to mark the starting and final positions of the moving object we have the distance traveled by it equal to: final position minus initial position
Distance Δx is the one traveled in the motion along a straight line, the x axis in this case. which has the same magnitude as Δx, but has a direction assigned to it, is called the displacement and it is a vector
If we assign direction to this distance Δx, then we must write it as and call it displacement. The equation defining displacement is analogical to Equation M1.1 and has the form
, , and are vectors.
Vector quantity: a quantity which has value and direction.
Scalar quantity: a quantity which has only value and no direction.
Examples of scalar quantities: time, mass, volume, energy.
Examples of vector quantities: displacement, velocity, force.
The positions of the moving object are represented in Equation M1.2 by vectors and . By definition vector is a vector with the magnitude xi, with an origin at position x=0, and with the direction along the x axis.
Motion depicted on Fig M1.1 is in the positive direction, because we define it as motion from position x1 to x2, and therefore the displacement is positive. Motion in the opposite direction (from x2 to x1) will result in displacement with a negative sign as x1 – x2 will be negative.
Remember: Reversing the direction of motion requires the changing of the sign of displacement describing this motion.
The displacement vector represents the overall effect of the motion, not the motion itself.
What does this mean? The displacement vector given by Equation M1.2 tells us that the moving object was moved from the position to , but it does not tell us what the path was along which the object traveled. This statement will be especially important and will become quite obvious in the analysis of the motion in two or three dimensions, as in motion along the straight line the displacement of an object is exactly along the path traveled by this object.
After displacement, the next parameter describing motion is velocity.
velocity: the ratio of the displacement to the time interval needed to “create” this displacement.
t1 and t2 are the times at which the moving object passes points x1 and x2. So the t2 – t1 is the time needed to travel the distance of the displacement.
Δ (the Greek letter delta): is used in all kinds of mathematical descriptions to denote the change in a quantity, the difference: final value – initial value.
Notice the arrows above symbols of velocity, positions, and displacement. These arrows are used to tell the reader that the symbols represent vectors. The other way of denoting vector quantities is to print symbols representing them in bold face type. In scientific literature the “bold method” is predominantly used, but in all kinds of textbooks arrows are more popularly used to denote vectors. The “arrows method” is more suggestive. It is also easer to distinguish such notation in text, rather than letters in bold. Not to mention that in handwriting using bold letters is out of the question. Therefore we will predominantly use arrows in this tutorial.
The Equation M1.3 is true only for motion with constant velocity. In other words – if the object is moving at the same pace during the time of “creating” the displacement. Of course, we are talking all the time about motion along a straight line which should be obvious from the fact that motion is along one axis (x axis in our case).
An example of such a motion may be a train on the straight part of a track, when it is moving with constant velocity or a car on a part of highway that is perfectly horizontal and straight.
In everyday life we often use the term speed for describing distance traveled per unit of time by a moving object. But speed is not the same as velocity and Physics distinguish them.
speed: the magnitude of velocity
When we looking at racing cars on a track we talk about the very high speeds of these cars. But notice that these cars do not move along a straight line, they do so only for short moments of time. Even if they move with constant speed their direction changes. So, even if the speed is constant, the velocity may not be. Speed is a commonly used parameter describing many types of physical phenomena. Take for example the speed of light, which has a precisely defined magnitude for the vacuum.