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

velocity: a measure of how fast and in what direction an object is moving, calculated by

displacement 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. We can

derive a formula that can capture aspects of the motion of the objects. We start by defining

velocity as the rate of change of position:

Re-arrange this equation and take an integral:

As discussed in the calculus section, the is the constant of integration. For this particular

equation, this is the object’s original position. So, we arrive at the formula:

Most of the time, this can be simplified in another way:

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.

Return to Equation (1), after the integral:

This looks like something we can graph. If we recall the equation of a line, we find

. Let's try to graph it:

In this case, would be our intercept (the object's starting position), and would be the slope

on this graph. The sign of the slope can tell us the direction in which the object is traveling.

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 the object’s speed to be

non-constant? We can derive new equations to use, of course! In order to do that, 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 and integrating, we can arrive at a slightly more helpful formula to use (but, in

reality, means the same thing:

Here, that constant is the initial velocity of the object:

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 us graph this new equation, and see what it 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:

We can make sure that the units for this are correct. We would be finding the area of the shaded

region, which is a trapezoid. The equation for the area of a trapezoid takes the form:

If we were to substitute in units for these bases and height, where the bases are and and the

height is :

Drop the constant, and simplify:

Thus, we can be more certain that the quantity we are finding is the displacement, as it has the

correct units. So, in equation form: .

To actually find the displacement, we can either find the area of the trapezoid, or find the area of

a triangle and add it to the area of a rectangle. However, this would be tedious and require some

algebra, and would not work for the more general case, where acceleration does not need to be

constant. We can, however, use calculus to find the area under the curve. We can do this by

using our acceleration equation:

But we know from previous equations that

, so substitute this in:

Here, is the initial position, so we arrive at the equation:

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 of a line tangent to the curve, here represented by that dashed line, 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.

If we want to find the slope of such a tangent line, then- as you may recall from calculus- we

take a derivative at that point:

We must derive one more equation before we begin using these formulas to solve problems. You

may have noticed that the previous two boxed equations both involve time. We can get rid of the

time-dependence by some clever algebra.

Start with:

And rearrange for :

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. In fact, we will only need to modify them slightly to talk about special topics,

especially if we have background knowledge in vectors.

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 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 these

formulas.

Freefall

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”

or “toward the sky” is positive. We represent gravity by the lowercase because it is the first

letter of the word gravity. (This is just one of many examples of our creative genius in naming

things.)

At any rate, to return to our example question: we now have three pieces of information.

, ,

We have an equation to use:

Assuming that the top of the building is “ ”,

So, the ball must have started at a height of .

Relative Motion

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.”

The equations of motion will apply for whatever observer is looking at the problem, and the

results will be solved the same way, and we should arrive at the same solution no matter what

reference frame we choose.

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.

If we were to pick a reference frame outside of both balls- say, the ground- it becomes apparent

what is actually happening. It does not make too much intuitive sense that the whole world be in

motion around one object, even if the equations of motion make it possible. So, for most

purposes, the preferred reference frame will be one that is stationary.

This process of finding alternate reference frames 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, because your actual speed is

the sum of your speed and the walkway’s speed. Or, when you step down with the down

escalator, you rapidly descend, for the same reason..

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:

If you have no velocity relative to the ground, then you do not move- you stay in place! This is

how treadmills actually work- if you match the treadmill’s speed, you do not move. As a result,

you can have an “infinitely long” track to use as an exercise tool, contained in a very small space

in your home.

It is important to remember that the equations of motion, as we have written them, apply in all

inertial reference frames, or points of view that are not accelerating. This means that our

observers will be sitting still, or moving with a constant speed. Any other reference frames will

be left to discuss with more advanced material.

With this, we now know enough “basics” of motion to begin getting into more complex

situations, such as moving in multiple dimensions.