Linear Motion
Now, we begin discussion of actual physics. We will begin by talking about the basics of motion, expanding and incorporating more general concepts of motion as we go. First, we will define a few terms:
position: the location of an object (with respect to a reference point)
reference point: a point in space from which we make our observations and calculations.
As an example, consider a number line:
0
This “0” would be considered our reference point. The darkened circle’s position is at -2, which is measured with respect to the 0. The gray circle’s position is +2, which is also measured with respect to the same position. If we consider the position of the dark circle our reference position, then the dark circle’s position is 0, and the gray circle’s position is +4. Being able to shift our reference point will prove useful, as often, calculations can become much simpler by choice of references.
This gives rise to the next two definitions:
distance: the sum of all changes in position
displacement: the change of position between starting and ending positions
This is a subtle difference. Consider that same number line above; if each circle started at 0 and moved to their new positions, the distance traveled for each would be 2. This is because distance is a scalar quantity; it only has magnitude. Their displacements, however, are different; the dark circle has a displacement of -2, while the gray circle has a displacement of +2. The sign of the displacement, in this case, describes a direction traveled, which makes displacement a vector quantity. If we were to represent these vectors with arrows, they would look like:
0
So, we have representations of how an object moves. If we incorporate a time period in which this object moves, we can find the object's speed or velocity:
speed: a measure of how fast an object is moving, calculated by distance over time
This, too, is a subtle difference in terms. Speed is a scalar quantity, and is often taken as the magnitude of the velocity vector. Velocities are vector quantities.
So, suppose the dark circle completes its journey in 0.5 s, and the gray circle takes 8 s. The speeds would be calculated using the formula:
So, if we assume 1 unit on the number line is 1 m,
Their velocities use displacement, which looks like:
So:
As before, the signs indicate the direction of travel.
If we were to rearrange those equations slightly, we can see them take the form:
Which may look like something we can graph... what was the equation of a line? Right- . Let's try to graph it:
These two equations work well when the speed of the object is constant, and the direction in which the object is traveling is also not changing. But what if we allow this to happen? We can derive new equations to use, of course! But first, we must define one more term. If velocity is changing, there must be some acceleration.
acceleration: a change of an object's velocity over time
This has its own equation:
Re-arranging, we can arrive at a slightly more helpful formula to use (but, in reality, means the same thing:
This equation describes how an object's velocity changes in response to an acceleration. For our purposes, accelerations will be constant, but in general, this does not always have to be the case.
Now... let's graph this, and see what this looks like!
Note how this looks like a line? Remember the general form of a line: . In this case, the intercept is our , and the slope is ! The sign of the slope can help us determine if the object is speeding up, or slowing down, or other aspects of its motion.
Now... suppose we wanted to find out the displacement the object experiences during some time interval. In order to do that, we can find the area under the curve of this graph:
From earlier:
Since , the equation becomes:
Which we can arrange to read:
This equation is very powerful; it describes the motion of an object that is accelerating or moving with constant speed. If we were to graph it, we would get a curve:
The slope a line tangent to the curve at every point would give you the speed at that point, called the instantaneous velocity. Instantaneous velocity is different from average velocity in that for average, we look at total displacement over total time. For instantaneous velocity, we can find the slope of the line (displacement over time) at that point in time.
We must derive one more equation before we begin using these formulas to solve problems. The previous two boxed equations both involve time. We can get rid of the time-dependence by some clever algebra.
Start with:
Plug this value into the other equation:
Begin expanding:
Multiple by on both sides:
Expand:
Begin combining terms:
Rearrange to the final form:
The three boxed equations are the main equations we use to describe motion in physics. They are powerful, in that they describe almost any kind of motion that we can deal with in the coming chapters. We will modify and derive other formulas as we go, but for now, we will use these.
For now, let us put them into a convenient table:
Here are some example problems that apply these formulas:
1) A car is traveling to the right at . It accelerates at a rate of for . What is the car’s final velocity?
We know: , , and , and
Based on this, use:
2) What distance did a track runner travel, if she had an initial velocity of , an
acceleration of , and completed her event in ?
, , (assume her initial position is )
3) If a bicyclist was able to travel , with an initial speed of and a final speed of , what was the cyclist’s acceleration? (assume the acceleration was constant.)
, ,
Obviously, these are simplified examples; there will be times when there will be multiple steps to a problem, or may involve solving two equations at once. Through class exercises and
homework, we will work on this process in order to become more comfortable with manipulating formulas.
Now, consider something new: motion in the vertical direction. Take this example problem:
A physics student dropped a ball, from rest, from the top of a tall building and measured the time it took to fall. If it took to reach the ground, how tall is the building?
It appears on its face, that we have not been given enough information to solve this problem. After all, we only know two things: initial velocity, and time. Consider: what is it that is making this object fall? Gravity! Gravity is what is pulling on, or accelerating, the ball. For reasons we will explain later, gravity near the surface of the Earth has a constant value: , where the negative indicates that gravity points toward the surface of the Earth and we assume that “up” is positive. We represent gravity by the lowercase because it is the first letter of the word gravity, and this is just one of many examples of our creative genius in naming things.
At any rate, to answer the question: we now have three pieces of information.
, ,
We have an equation to use:
Assuming that the top of the building is “ ”,
The negative should make sense; remember, “0” was at the top of the building, and the ball fell from there. If we repeated the calculation with the ground being “0”, we would find it was initially at . Since the question just asked for the height of the building, we can leave the negative sign off and just report the magnitude.
There is one final thing to consider: that of relative motion. We have skirted around this topic so far, but before we go further, we must address it. Relative motion simply means that motion is relative; that from a point of view, the equations that govern the motion of an object may need to be modified slightly to describe it from a different point of view. These “points of view” are called “reference frames.”
As an example, consider two balls: one rolling toward the other, the other stationary.
From the white ball’s perspective, the black ball is moving toward it. However, looking from the black ball’s perspective, the white ball appears to be the ball in motion, with a speed of – ( to the left). From the black ball’s point of view, it thinks the world is in motion around it! Both of these frames of reference are accurately described, or correct. It is just that when we do
problems, we must be careful about picking one frame of reference, and sticking to it.
This also works for objects that are in motion. Consider a motor boat traveling with a current:
The boat is going downstream with a velocity , and the river is flowing with a speed . From the perspective of someone on the bank of the river, the boat actually has a total velocity of
. The same could be said about moving walkways in airports; when you move
with a moving walkway, you can rapidly speed along to your gate. Or, when you step down with the down escalator, you rapidly descend.
What if you moved in the opposite direction the moving walkway was designed to go? Suppose the walkway had a speed of relative to the ground, and you were walking in the opposite direction with a speed of , again, relative to the ground. Your total velocity would be:
